The Achievement does not mention you have to defeat the boss of the current world you attempt the achievement in, or subsequently that you have to enter one of the following areas.

One could speculate that the Boss of an area belongs to stage three,
as that would explain why the achievement only unlocks after transitioning to the next area.
This was suggested by u/A_Human_Space_Heater on the r/ShovelKnight subreddit.

This was information I was missing too and only found out through trial and error as there isnt much on the achievement out there.

I will explain the method I used which seems to be the most common from what I can tell.

First off, you want to have the Streamline Mail unlocked and equip it,


you will be unable to access side paths but it also has two effects that play to our advantage:

1. No Golden Gears
   
   • this is good because you wont be able to pickup gems from Drill Knights capsule,
     but still get a free accessory after every stage.
2. Most monsters dont drop gems upon defeat
   
   • there are tons of situations where enemies will be in your way, killing them might end the run due to their gems, letting yourself get hit takes away valuable HP which we will need for Spore Knight.

1. Reach Stage Selection for Stage 3
   Optional: Picking up a Mobile Gear from the capsule if given the option prior to stage 3
   
   
2. If you have a path with a sign depicting a sideways drill, go for it, if not, restart until you get one.
   
   
   
   Depicted above are the symbols in question.
   
   
3. Remember how many gems you currently hold.
   
   
4. Let the big drills eliminate gems you cant get past, you sometimes have to be a little patient.
   If you have a Mobile Gear Relic you can also use it to clear out gems, but it is risky.
   
   If you have reached the end and have more gems than when you started stage 3, start over.
   
   
5. Pick a combat relic or health before fighting the boss, whatever makes more sense at the moment.
   
   
6. Defeat Spore Knight, you will have 4 HP most of your attempts at best, 5 if you are lucky with the capsules, so make sure you are familiar with their movement and attack pattern.
   
   
7. Enter Smeltworks or Secret Fountain.
   
   
8. Achievement should unlock during the transition animation.

This is how I myself unlocked the achievement on 09.07.2024, (DD.MM.YYYY)
on game Version 1.1.5.

I hope this guide was helpful to you.