In general when talking about moving to flats, I would start with a classic, industry-standard set like LaBella Deep Talkin's, Pyramids, or Rotosound black tapes (88s).
For what you describe, though, I would try D'Addario Chromes or Rotosound 77s. They are a bit more subdued in the bassiness department, and have a bit more metallic clank on tap if you want it (e.g. Steve Harris has used RS 77s since the '70s). Chromes are a more modern "slightly scooped" sound, while RS77s are more classic sounding in the low midrange (but not as much so as LaBellas or Pyramids). Personally, I dislike the anemic midrange of Chromes. It's just a little bit too weak for me. They just don't sound "big" – like the classic '50s and '60s bass sounds that are my 'home base" tone...and I really don't like the way they react when you dial treble out of them. They become very indistinct and weak. But that withdrawn midrange of the Chromes might be just the ticket for what you want. The RS 77s are more what I would call "balanced" or "flat across the board" in terms of e.q., not mid-scooped.
But, all that being said, as long as you have excellent e.q. controls on your rig, and know how to use them, the primary concern is feel, IMO/IME. Do you want a string that "rolls" and "springs" under your fingers, or a string that keeps its position when you touch it? One with a polished finish, or a bit of "grit" on its surface? One with gaps between the windings, or a totally continuous feel, such that you can't even tell it has windings? High tension, medium tension, low tension? Etc.
I want high tension, no gaps between the windings, smooth but not slippery feel, and no springiness or roll on the right hand...and the inherent tone to be warm, "plucky," and classic. LaBella Deep Talkin' .110s are my go-to set, used on all my go-to basses. I have other sets on other basses, however, covering a decent range of various flat wound feels (and tones). I don't go as far as TIs for loose feel, or as far as Chromes for modern tone. I pretty much use nothing but flats, and always have but I have some basses with half rounds and even some with rounds.