Yes.
Several of the fire ant species have a nearly or indeed all-black color variant, S. xyloni, S. geminata, S. invicta (this last only in a few locations in western Mato Grosso, Brazil).
There are also some partly or all-black small Solenopsis, but the great majority are brown, red or yellowish.
On the other hand, in Australia, Asia and Africa, there are many reddish Monomorium species, including some that look a LOT like little Solenopsis, until you examine their antennae.