Before anything else, ensure that you have at least one full stack of coal (~20) in the drill's storage so your Drillship doesn't run out while it's traveling from site to site without you. If your fuel supply is critical, surface at a site near the perimeter of the island and mine up a nearby coal deposit so you'll have an ample reserve, as you won't be putting any more in storage until step 10. If you have a Pilot Seat, you can also look for more coal underground as you travel, but you won't be able to do so again once you get started, as you'll mainly be relying on the autopilot to control the Drillship while you wander around on foot, which won't pick up any resources it comes across.

First off, go to the Safe Grounds, as this is where the whole thing starts. While your Drillship is travelling, take this time to heal-up, it'll help you in the long run. If your time is low, or you need extra time to make a medkit, use the up/down arrows under the right-hand screen to switch the Target Selector from "surface" to "underground" so that the Drillship will stop instead of surfacing. If you prefer a Pilot Seat, or don't have access to a Target Selector, you can instead use the time spent traveling to search for underground coal deposits, and can simply stop nearby once you get there.* Another recommended course of action is to wait until the next eruption, so that you'll have the full amount of time to collect as much coal as possible.

*Hint: You don't need to be on top of the Landing Site icon to be able to surface. Just get close enough for the "Go to surface" button to light up, and you can surface at any time.

Clear out your inventory so it's completely empty. If you leave anything in your inventory, it will reduce the number of slots you have for coal, which in turn will increase the number of trips you'll have to make, and in turn, the number of times you'll have to wait out the eruption. You'll still keep your wrench and pickaxe, you just won't have anything else.

Go down to the beach, starting near the house on the north end. Why the house? Because it's at the western extent of the area you will be traversing. Though if you see any coal spots to the north of the house, you can start there instead.

Send your Drillship underground remotely with G, because you'll be away from it for a long time, and by sending it underground without you, you won't have to worry about it later. Plus, it will arrive sooner when the time comes to recall it. Likewise, it's advised that you sprint as much as possible, or you might find yourself in the path of an oncoming ash cloud.

Follow the beach around the south side of the island, and mine up every coal deposit you can find. If you have room for it, you can go past the sub and get as much of the coal on the hill that you can hold as well, but be mindful of the timer, and be ready to scram when your time gets low.

Enter the sub and drop off your coal in the storage module, you've probably collected a lot of coal by now, and you'll need a clear inventory for the next area. Don't bother trying to recall the Drillship, as it will take too long unless you call it from a distance so that it surfaces at The Harbor as you approach. Additionally, there's a chance it could be destroyed by the eruption if you call it too late, so it's safer to just leave it underground.

If your time is low, you can wait out the eruption inside. If you don't have them yet, you can also get two of the secret achievements in here as well: Simply close the door leading to the captain's room, and the door in the sail that leads outside (not the floor hatch, the actual door).

When you're ready, exit the sub, head up the hill to the east, and get all of the coal up there that you missed before, then continue east and mine up all of the coal that you can from that beach.

Once you get to the northeastern end of the beach, duck into the coal mine and get as much coal as you can scrounge up (if you somehow fill up your inventory a second time, you can safely ignore any leftovers, as you should have plenty by now), then call your Drillship to Boulder Shores and put that coal in storage. If you have enough time, you can continue with steps 10-11, but once you hit the 1:00 mark, you should drop everything and make a beeline for either the coal mine, or one of the towns.
(Note to Fire: This picture needs to be redone, as the landing sites were given actual names instead of just numbers. It is now "Boulder Shores")

Once you're finished with the mine, head down the other side of the cliff you just climbed up, and go onto that beach, mining your way towards the Hunter's House. Make sure to avoid any errant COGs as you near it, as you have no real means of combating them at the moment, but if a fight is imminent, your only real option is to recall the Drillship and hope the turrets can take them down before you run out of health, unless you know for an absolute fact that you can defeat them on your own with nothing but a pickaxe.

While you're in the area, head into the Hunter's House area and grab the destroyed modules if you have room, but be mindful of any lingering COGs. If there's a COG Drillship on the Craftsmen's Site, avoid it at all costs. If there's a COG Drillship on Boulder Shores as well, you'll need to run to another site and recall your Drillship from there. If you had to 'axe any COGs along the way, you can search them too, but don't bother getting anything unless you need a medkit, anything more will just take up space.

After you make it through the Hunter's House, don't stop yet, that's simply the 2/3 mark of your journey. Keep going until you reach a small cliff that's at about the same "height" as the Old Refinery Village on the map. If you haven't maxed out your inventory by the time you reach this area, you've either passed some coal spots, or are playing a game with Deposit Richness set to a low value. Once this is done, backtrack to the Hunter's House and call your Drillship one last time. Again; if the nearest site is blocked, you'll need to run to another site and call it from there. Wait for your Drillship to deploy, then get inside, making sure not to let any COGs that followed you make their way into your turrets' blind spots.

If you're low on time, or you have one or more COGs that made it into your blind spots, send your Drillship underground again. It is recommended to use G for this purpose unless your entrance is close enough to either the Target Selector, Pilot Seat, or a Lever Stand to negate the three-second buffer that the recall button has.

Put your newly obtained coal into storage. If you're the kind of person that likes to give each resource it's own storage module, find the storage module you assigned for coal and put it into that container directly. If you prefer to use the online inventory exclusively (and thus don't care where anything goes), you can just throw all of it into the first storage module you come across.

When it's safe to do so, take your Drillship back to The Harbor and pick up the coal you left in the sub, then return to the Drillship and put that coal in your storage as well. If you had to hide from the eruption at any time, it might be worth it to go through the mine and across the beach to get whatever coal respawned, just make sure you have enough room for it all.

If you followed this guide, and still want more, wait for the ores to refresh, then start back at the beginning when you're ready. If you had to hide from the eruption at any time, you can start again immediately if you like, as the ores on the western beach should've refreshed by now. If you don't have enough time, set the target selector to Underground so you'll be ready to start as soon as the ores do refresh, or hop in the Pilot Seat and drive there manually so you can pick up any coal along the route. Keep in mind that they might not refresh automatically at the end of the eruption cycle, so you may need to spend a bit of time looking through the periscope. Though if you do need more, and have been using the Pilot Seat, you should have collected more than enough coal to cruise around underground in search of any deposits while you wait for the surface deposits to refresh. Even if the surface deposits have refreshed, using the pilot seat to search for underground deposits may allow you to run out the eruption timer, so you'll have the full amount of time to work on the surface deposits.

Note: Once you gain access to the Mining Drone, this guide will no longer be needed, as you can simply set one such drone to only collect coal, or set all of them to collect coal before anything else. Just be sure that if you choose the latter, you remember to remove coal from the active filters before you get too much coal.

If this guide helped you with your coal problems, or allowed you to finally get that one achievement you've been searching for, then I'm glad I could help. If you've gone off the rails and are now dead, or somehow managed to wreck your Drillship... Well~ that's what the autosave is for. (The creators of this guide are not responsible for any damages, real or virtual, that may result from a failure to use this guide to the best of your abilities.) If you have any questions, don't hesitate to post them in the comments, but do note that any non-constructive criticism or flamebait, as well as non-English, poorly written, or otherwise sub-standard or offensive comments will be deleted without hesitation.

Credits:
Gofastmike: Initial concept, guide creator, and all text.
TheFireDragon: All images within the guide itself. (WIP)
Volcanoids dev team: Image used as guide title card.