The word POST has several distinct meanings.
Your two examples are quite different.
Post-haste has nothing to do with after; it means with great speed (probably related to the official conveyance of parcels, letters, etc. ... while Americans send it, or used to send it, by mail, the British send it by post).
In post-game, POST is a prefix meaning after in time or order.
There's also POST no bills, the POSTS that are set upright in the ground, the finish POST in a race, POSTING a letter, keeping someone POSTED, entries that are POSTED in a ledger, the first or last POST (bugle-calls), a military POST....