My comic Errant:
Is a very hard reboot of my old comic, Fan Dan Go:
Since it's been a number of years, and FDG was on sites like Smackjeeves, Drunkduck and a self-hosted site, while Errant is on Tapas, I just left the old version there. It's a nice little artefact and I'm proud of how hard I worked in the past on this old version from my early 20s, even though the new one is so much better.
I think if the tone is going to be distinctly different, make a new series with a new name, even if it shares some characters and plot elements. This isn't all that unusual, you can see it in John Allison's work:
http://www.scarygoround.com/
Scary Go round is a sort of reboot/sequel to Bobbins, but shares some characters, and then Bad Machinery is the same to Scary Go Round etc.
There's also a similar relationship but with hard rebooting of the same characters into comics with different tone and premise in David Willis' work, which goes: Roomies! > It's Walky! > Joyce and Walky > Shortpacked > Dumbing of Age
https://www.shortpacked.com/about
They all share characters, but only the first three have sort of continuity.