To help jog my memory, I went looking for lists of one-hit wonders.
From those lists, here are some that I can vouch for:
Men Without Hats - found a couple of their albums in the discount bins, back in the day, and quite liked them, though more in a “guilty pleasure” way than a “these guys should have been way bigger” way.
The Proclaimers - ditto
Semisonic - I like them a lot, and I love the earlier band whose ashes they arose from (Trip Shakespeare).
Fountains of Wayne - As @Sam_Stone said, a large catalog of truly great songs. Their Utopia Parkway was on my list of great power pop albums before I ever heard of “Stacy’s Mom.”
Eddy Grant - his most successful album was the one with “Electric Avenue,” but I enjoyed the followup, Going For Broke.
Madness - I’m not sure they even count as a one-hit wonder in the U.S., but they’re way more than that in the U.K. Lots of wonderful songs.
Timbuk 3 - Eden Alley is an interesting album.
Chris DeBurgh - Like many artists here, I know more than just his one hit thanks to my habit of browsing the discount bins during the 80s and 90s. Flying Colours is a good solid album.
Til Tuesday - Their third album, Everything’s Different Now, is a pop gem.
XTC - Do they really count as a one-hit wonder? I wouldn’t have thought so. They’re a great band, at any rate.
General Public - Hand To Mouth was another good album I serendipitously found in a discount bin.
Midnight Oil - Yet another band whose one-hit wonder status is US-centric. Earth and Sun and Moon is my favorite album of theirs.