The Other Ones was the Grateful Dead after Jerry Garcia’s death.
How do you figure that, given that Krist Novoselic has never been a member of Foo Fighters? (He played on one song of theirs as a guest musician).
Bruce Baker Moore was Cream with Gary Moore standing in for Eric Clapton. Not half bad, though.
Ok 
And Tones on Tail, minus Murphy and David J, plus a roadie.
She left for 10 years, but now Hook is gone.
Scott Weiland (also known for Stone Temple Pilots) is/was the singer for Velvet Revolver. But not sure he is in the band any more.
Also in Velvet Revolver:
Slash
Dave Kushner
Duff McKagan
Matt Sorum
Slash, Duff, and Matt were from GnR. Former GnR guitarist Izzy Stradlin is missing from Velvet Revolver though. So they are more like 3/5 of GnR. Even less if you’re counting Dizzy Reed. GnR had a complicated history.
Anderson, Bruford, Wakeman & Howe was Yes without Chris Squire.
We are the Fallen is Evanescence with Carly Smithson instead of Amy Lee.
Mickey & The Hartbeats were The Grateful Dead without Bill Kruetzman, the better of their two drummers. And I want to amend that to make sure that everyone knows I think Mickey Hart is a wonderful drummer. It’s just that Kruetzman has that extra special something.
I apologize but this is minus two members not one, but does fit OP case overall. Arcadia was a short lived side project which was Duran Duran minus two members.
The Grandmothers were The Mothers of Invention without Frank Zappa.
The Jacksons were the Jackson 5 without Michael.
Off-topic, but OMG that article may be the saddest band history ever.
Where did you get this bizarre idea? Michael Jackson was certainly a member of The Jacksons, and in fact he sang lead on their biggest hits: “Enjoy Yourself,” “Shake Your Body (Down to the Ground),” “Can You Feel It?” and “State of Shock.”
You could make a case that The Jacksons were the Jackson 5 without Jermaine Jackson, although he eventually did join the group some years down the road.
ALL was the band the Descendents became whenever Milo was busy in college or working as a biologist.
Yes a band with a very sad story.
2/3s of Blink-182 formed a side project without the bassist called Box Car Racer although the bassist played on a couple of the tracks anyway. Technically a different band, but only barely.
Not quite one member but, the (original) Wiggles were a fair chunk of an Aussie pub band called The Cockroaches.
Annie Lennox is The Eurythmics without David Stewart.
I was going to say the same thing about Geronimo Black.
The first Long Hello album was recorded by the ex-members of Van Der Graaf Generator after they split in 1972* (plus Nic Potter, who had left earlier) without lead singer/songwriter Peter Hammill.
The line up was David Jackson, Hugh Banton, Guy Evans and Nic Potter although I don’t think they ever did come up with a name for themselves.
The subsequent Long Hello albums were projects involving 2 or more of them…
- the same line-up reformed 3 years later
The backing band for Bob Weir’s 1972 solo album, Ace, fits the OP: Jerry, Phil, Keith, Donna, and Billy. It fits the OP because Mickey wasn’t part of the band at the time but Pig was—one member missing.
OT: Did you catch Billy’s little impromptu performance a couple days ago?