Make sure you have at least unlocked Tier 2 contracts so you can reset your contract on-demand.

Reset your contract to Tier 1 with a 200 quota requirement. After every contract, reset your contract back to 200 by selecting another tier of contract, then re-selecting Tier 1.

Equip a weapon and an Echo Lens, and a Boombox if you want that you can leave outside of the ship to passively heal you when you come back.
(You can also buy a Jetpack as it increases your movement speed, but I don't use it because it can break your FOV).

Go to the area with the highest density of access crates and use the Echo Lens to search level 0 crates for the code to level 1.
For example, in Port Coral, this area would be straight forward as you exit the ship: Under the water and inside the facility that you can take an elevator to enter. It will take some experience before you have enough knowledge of where to go on every map to be the most efficient.

Do the same for level 1 and level 2 until you get the level 3 code. You can then use the Echo Lens to go to all of the level 3 doors until you retrieve the egg.

Remember to deposit the egg in the Egg Drop Off BEFORE you leave the planet.

While you're searching for codes, pick up loot and hold it in your action bar and deposit it whenever you go back to your ship to input a code. In this way, you will always have more than enough quota to fill the contract. (Just remember that creature corpses are always more valuable than things you find inside the crates themselves).