In the warm-but-not-so-woolly department, I used to have a fretted 4-string Epiphone El Capitan.
IME, the acoustic sound of an ABG isn't all that important, other than for the effect it has on the plugged-in tone. Between the short decay and the piezo setup on the Epi, I got a big sound with a powerful punch and bite. It sounded halfway between a hollow-body fretted EBG and an upright. Kind of like a Beatle bass, but with big, hairy balls...
I think ABG's get a bum rap. When properly amplified, they have an interesting and unique tone that's useful for much more than woodshedding and living room jams. I used mine successfully for everything from Rockabilly to Alt-Pop to up-tempo church music. Various bands I was playing with were upset with me when I sold it, but I got tired of carrying two basses. The short decay and punch that made it such a good instrument for up-tempo stuff were counter-productive on ballads, and it seemed like no matter what the setting, there were always songs that required my 5-string solidbody.
In the end, the flexibility of my fiver won out, but in a perfect world where I had more money and a stronger back, I'd still have an ABG. Ironically, I just picked up a Squire VM fretless, hoping that it would fill a similar sonic space, in a more portable package. Judging from my first gig, it works, but doesn't hit the spot quite as well as the El Capitan did.