Rehashing can be done in the background, but then you had added code complexity, you'll be doing significantly more I/O during the rehashing (did you think about rehashing when sizing your machines?) and in pathological cases, you can run out of hash table space before the resize is complete. Now what?
1: Allocate a new table of size at least 2X. 2: All accesses to the table hit both the old and new tables. 3: Inserts only go in new. 4: Any change (insert, update, delete) also moves 2 other elements.
You can prevent a O(n) rescan at 4 by keeping a single persistent cursor that scans through the old table from top to bottom. There, still O(1) throughout (C may go up by 2-3 but that's still small). And only uses 1.5x memory.
Not saying that's what postgres does here (it uses WAL, so probably not exactly).