Billboard says:
“New Orleans is Sinking” (the song I was thinking of) reached #30 on the “Mainstream Rock” chart. I thought that “Blow at High Dough” might have made it too, but it didn’t.
“Courage” reached #16 on both the “Mainstream Rock” and “Modern Rock” charts. That’s news to me, because I have the CD, and have never heard any of the songs from it anywhere else. (I would have picked “Locked in the Trunk of a Car” or “At the Hundredth Meridian” as a more likely hit.)
“Poets” blipped onto the “Mainstream Rock” chart for one week at #39. I’ve never heard it.
A year later this act got to #27 in the U.S. with “N.Y., You Got Me Dancing”, which I remember as being a reasonable good pop song.
And, in response to the references to Tommy Tutone, a year-and-a-half before “867-5309/Jenny” they had a #38 hit, “Angel Say No”. Because of this earlier minor hit, I remember thinking, at the time “Jenny” started to be played, “Boy, I never thought I was going to ever hear something by this group again” due to their earlier, somewhat forgettable song.
OK, geezer checking in:
60’s R&B
Sallie Go Round The Roses – Jaynette
The Love of My Man – Theola Kilgore
When A Man Loves a Woman – Percy Sledge
You Better Move On – Arthur Alexander
Also
Total Eclipse of the Heart – Bonnie Tyler
You Keep Me Hanging On – Mary McCaslin (cover)
Never heard of any of these?
Go ask your Grandparents.
Hey Old Goat I’m a child of the 50’s/60’s myself. Every song you mentioned played in my head as I ran down the list.
Here’s a song you may remember, with an organ-driven hook that could bang around in your head for days - Green-Eyed Lady by Sugarloaf, a band from Boulder, Colorado that nobody ever heard from again.
Sugarloaf also had “Don’t Call Us, We’ll Call You.”
The organ line from “Green Eyed Lady” was later sampled in the Eerie Splendor remix of “Doin’ Time” by Sublime.
I’m no grandparent, but I’m old enough to know that Bonnie Tyler also had some success with “It’s a Heartache,” so I don’t think she qualifies as a one-hit wonder.
I think we can find a lot in those fluid bands that were created by producers to record one song, but whether or not they’re any good (as the OP wants) is open to debate. “Love Grows (Where My Rosemary Goes)” by Edison Lighthouse is a good example. According to the band’s Wikipedia page, the lead singer was active in more than one band–indeed, he led other one-hit wonders:
As for “good, even great” one hit wonders, I’l toss out a few that I feel fit the criteria. They’re instrumentals, but the first two were Top Ten hits; the third hit #11:
Joy, by Apollo 100 (hit # 4 in February 1972): A synthesized version of Bach’s “Jesu Joy of Man’s Desiring.”
Popcorn, by Hot Butter (hit #9 in October, 1972): More synthesizers. This was an original composition, and proof that they were here to stay.
Amazing Grace, by the Royal Scots Dragoon Guards (hit #11 in July 1972): Yes, a military bagpipe band playing a church hymn hit the pop charts once. Never again, but whodathunkit?
I WANNA ROCK! I WANNA ROCK! I WANNA ROCK! I WANT TO ROCK!
“Holding out for a Hero” (from the Footloose soundtrack) made #34 in the US. #1 in the UK.
The best holiday one hit wonder: “Grandma Got Run Over By A Reindeer” by Patsy & Elmo.
By that standard, almost anything is tolerable.
Mississippi Queen by Mountain
Norman Greenbaum’s “Spirit in the Sky.” A huge hit for him and also for Doctor & The Medics. And a cool song.
“Right Place, Wrong Time” (YouTube clip) - Dr. John
Ahhh…a song about multiple orgasms by a former porn star.
Good times, those.
Gad, I loved that song when I was a kid. As well as being in Footloose, it was also the opening music for short-lived 80s TV series “Cover Up”. (Short-lived because Jon-Erik Hexum accidentally killed himself with a blank-loaded gun on the set. Ouch.)
I can’t believe this is his only song that charted. Yowza. But I agree with the OP that he is not a one-hit wonder by any non-technical definition of the phrase.
Not being religious helps. As a religious person, I hear “What if God was one of us?” and think “He was one of us, and we chose to torture and kill Him” and then switch the station. I admit this is an outlier reaction! 
I’m also a hater of “One of Us.” Yet, I bought the album. Why? Because every other song on the album is better and a couple (“Ladder,” “St. Teresa”) are transcendent.
“Inna Godda Da Vida”- Iron Butterfly.
“You Keep Me Hanging On”- Vanilla Fudge version.
“Driver’s Seat”- Sniff and the Tears
If any of these groups had any other “hits” I am not aware of them.
I tried to see Zevon in Seattle in the early 90s. Sadly his concert was cancelled for lack of interest, only 20 tickets were sold 
I was personally raving and recommending the concert to everyone I knew beforehand also, and everyone just said “who?”.
I really regret never getting the chance to see him, he really didn’t get the respect or recognition he deserved.