Why isn't the previous Battle Pass season in the Archive? When will it appear?

A season isn't added to the Archive the moment it ends — it appears one Battle Pass later. The season that just finished becomes available in the Archive only when the Battle Pass currently running comes to an end.

For example: while the Pirate Battle Pass is running, the season before it isn't in the Archive yet. It appears there once the pirate season ends.

If you bought that Battle Pass, it will show up in your Archive as already owned — you don't pay for it again, and you can keep unlocking the rewards you didn't finish.

If you didn't buy it, it becomes available for purchase in the Archive at the same moment.

Either way nothing is lost — the skins from that season stay available through the Archive.