There have been a lot of people that have been researching and asking online about how to get the Atlas Passes V2 and V3, but none have been much help or are inaccurate, so I will post how I received them as an answer.
To gain the Atlas passes V2 and V3 you must deploy a signal scanner (orange rays of light shining up to the sky) and upon interacting with it you then need to select colonial outpost. At this point it will give you a waypoint called either "Manufacturing Facility" or "Operations Centre". The Operations Centre is the only one that can potentially have the Atlas Passes (as one might expect since they are the rarer of the two that show up on most planets).
Once you arrive you will need to destroy the door to gain access to the building (if you go straight inside the sentinels will leave you alone). You then need to access the database and select the right answer. If you do not get either of the Atlas Passes do not be worried you are doing it correct, you just need to keep doing these steps until you find the correct blueprints.
TIPS: When I found them it gave me two options and they were V3 was the bottom option and the V2 was the top option (they are not the same, you will have to carry on until you find the other version, unless you are lucky enough to get V3 on your first try, which opens ALL Atlas doors, but not V1 Atlas debris canisters, so keep your V1 pass if you want to continue opening those).
That's how you gain access to the Atlas Pass V2 and V3.
Polo will eventually give you Atlas Pass v2 and v3, but only after many tasks. The 12th reward Polo gives is Atlas Pass v2 in exchange for having Ships Destroyed level 8. The 14th reward is Atlas Pass v3 in exchange for Extreme Survival 7.
In addition people report randomly finding the v2 or v3 passes in Operations Centers in Korvax systems, after choosing to Silicate and getting a blue blueprint reward.