One of the big things that helps with Jack is how quickly it makes just about any hand a consistent $5 or better. Take something like Lookout. Normally Lookout is utter crap as it nerfs your current hand and also prevents you from buying big cost cards. Jack can draw back the "lost" card, gets a better draw from Lookout's sifting, and adds silvers so you quickly hit $5 and $6 even with a Lookout/money hand. Virtually all non-terminal trashing works exceedingly well with Jack. Or take some of the weakest trashing, like Trade Route. Normally, Trade route is something like a -$1 to -$3 when you play it (e.g. could have been a silver, gives nothing until someone buys a green, and kills a copper), but with Jack and villages, you can trash out draw back up; as a bonus, you can power up your Trade route with a spare estate buy ... and kill the estate with the Jack who otherwise wouldn't trash a card. Jack allows you to mix in a lot of cares that tank the current hand's buying power without spending so long rebuilding your deck.
Weak sifters also get a huge leg up with Jack. Cellar typically costs 1 card in hand size and gives you poor odds of improving your hand. Warehouse gives you much better odds of improving your draw, but still costs a card. Storehouse, with a village, works great with Jack - sift for the Jack, discard for +coin, draw a fresh hand with Jack.
This is absolutely huge in strong mirror engine settings. Normally there is a tension between getting good hands now (say open Silver/Silver) during the big rush on Minions, Hunting Parties, Torturers, Wharves, etc. and having a thinner, better deck later (say using light trashing). Jack makes it a lot more viable to use really weak trashing and sifting and still hit $5. Partly this is because Jack really moves the average card value in the deck near $1.6 really quickly, and partly this is because its nature as an after the fact "Moat" allows it to offset card penalties (like smaller hands thanks to Lookout or top decked coppers thanks to Mandarin).