I first thought about Chapel, but Chapel may work not as well with Gardens.
Sentry, however, is usable in any kingdom. It is nonterminal, can sift, trash, inspect the deck and change the deck order. Other very versatile cards include Wishing Well, Vagrant, Necromancer and Ironmonger.
Vagrant: Strong trasher in kingdom, source of VP tokens as a viable non Victory-card strategy, no relevant attacks, other
cost strictly better than plain cantrip (such as pawn)
I couldn't think of any reason I wouldn't go for any of the other cards on that list. The below are all from the wiki.
Ironmonger: Heavy cursing
Vagrant (again): If draw to X is the clearly dominant strategy for the kingdom, but there's weak trashing.
Necromancer: Rogue or Graverobber in kingdom
Sentry: Stronger trashing (although I'm not sure about that. Sentry might still be pretty good for the deck inspection/sifting). It also said it's not great with big money. You might still want one, I think. Maybe if both conditions are true you don't go for it? But if there's stronger trashing, then you don't go big money usually.
Wishing Well: The wiki didn't offer any reason here, other than "Silver, which is usually superior". I'm not sure about that. I mean, it's probably true in big money, but even then you might want at least one Wishing Well.
I wasn't sure about the wiki's answers for antisynergies with Sentry and Wishing Well. At least, none of them were antisynergetic enough for me to be satisfied saying I won't go for them even once in that situation. Am I wrong; are these reasons enough? Are these the cards that are good in any situation? Is there a different reason not to go for them?