Chromie time burning crusade. Turn off XP. Farm rares in eastern kingdoms / kalimdor. These don't proc tertiary stats or sockets, but are very fast for a base set of 344s.
I am currently doing the level 59 version of this - I did Chromie Time burning crusade, turned off XP and farmed kalimdor / eastern kingdoms rares for 320 iLvl gear (old enchants now work up to iLvl 320). This lets me farm 320 boe boots to enchant with 10% speed (replace all those old legion iLvl 50 boots with 10% speed).
The downside to this strategy is the lack of sockets/tertiary stats in these old world zones.
I am now making a list of shadowlands and BFA BoE greens that CAN proc a socket and/or tertiary stats at iLvl 320 (344 for you). This will be much harder to farm, but more rewarding / maybe even sell for good prices.
Edit: If anyone has a level 56 - can you please kill an old world rare and see if it drops iLvl 320? All the 320s I have farmed at level 59 say require level 56.. but wowhead gear iLvl scaler suggests otherwise :S
