Hotlegs (1 top 40 hit “Neanderthal Man”) > 10cc (several hits)
John Cougar > John Cougar Mellencamp > John Mellencamp (top 40s hits as all three; all songs are now billed as by John Mellencamp).
Also
The Wonder Who? <> The Four Seasons. For contractual reasons, the Four Seasons released several singles as The Wonder Who? One (“Don’t Think Twice, It’s All Right”) made it to #12
Bonzo Dog Doo-Dah Band > Bonzo Dog Band. Though their first album didn’t spawn any hits, they were very well known in the UK due to their TV appearances, and it contains their best-known song. They changed their name for album #2 and had a UK single hit with it.
Fiddler’s Dram had one hit single, The Day We Went To Bangor, and subsequently became The Oyster Ceilidh Band, then The Oyster Band, and most recently Oysterband.
Fairport Convention released at least one album as Fairport.
There’s Will Oldham’s various Palace incarnations prior to Bonnie “Prince” Billy, including Palace Brothers, Palace Music, and Palace Song.
Fields Of The Nephilim changed their name to Nefilim and back again, although Carl McCoy was the only constant member of the various incarnations.
Ridiculous epic metal band Rhapsody changed their name to Rhapsody of Fire after an Australian girl group claimed they owned the name. This prompted the marginally less ridiculous parody band Nanowar to change their name to Nanowar of Steel.
Brian Setzer did indeed go on to a long (still going) and successful career without the Stray Cats, but I’m not sure he took the name with him. To the best of my knowledge, he never performed under the Stray Cats name except for “reunion” shows, with Rocker and Phantom (or Drucker and McDonnell, if they had to show ID). So I’m not sure it’s accurate to say that Stray Cats became Phantom, Rocker and Slick. Some membership overlap, but nobody was claiming that they were the same band.
But there really was a name change. Before hitting it big as the Stray Cats, they were the Tomcats, a Long Island bar band. A really, really good Long Island bar band. And they occasionally went by Brian and the Tomcats.
This one doesn’t quit meet your criterion, but darn it all, I’m going to mention it anyway. In England, a group called the Mark Four released four non-charting singles in 1963-66. They then replaced their bassist and changed their name to the Creation, and had a top 40 hit in England with “Painter Man”.
Pete Frame’s Rock Family Trees probably has some good examples.
Jethro Tull played under the names ‘Navy Blues’ and ‘Bag of Blues’ (among others more obscure).
They cut a single called ‘Sunshine Day’ under the name Jethro Tull, but the name of the group on the single was mislabeled as ‘Jethro Toe’.
Yeah, they never changed the name of the album in its US releases (my copies say “No Answer” on them, too), even after it was realized that it was in error. UK releases of the album have always simply been entitled “The Electric Light Orchestra”.
Roy Wood was in ELO for the first album (and, theoretically, was still in the “lead”). He departed after that first album, and formed a separate group, while Lynne kept the ELO name.
I was shocked that no one had mentioned
Jefferson Airplane > Jefferson Starship > Starship
but then I read the OP again.
But there is also:
Majesty > Dream Theater
had to change their name because there was another Majesty when they were signed to a label
Legacy > Testament
had to change their name because there was another Legacy when they were signed to a label
Rhapsody > Rhapsody of Fire
one of the most popular European power metal acts for a long time under their original name, but then Rhapsody the online music service caught up with them
Shaman (Finland) > Korpiklaani
they had been around under their original name for a while and had released two albums, but then a Brazilian band featuring former members of Angra formed with the same name, and rather than going through a legal headache they just changed their name to something that nobody would think to use
Shaman (Brazil) > Shaaman > Shaman
former members of Angra formed under this name and released one album, but then learned that there was a band from Finland that had been around longer with the same name, and rather than going through a legal headache they added an extra “a” when they released their second album…then they realized the other Shaman went and changed their name anyway so they changed it back