Muddy Gap or Lander are your choices from last time I went through. If I were on the West Side I would go to Lander and get some grub, east side well then Muddy Gap is definitely closer. Muddy Gap is expensive, but it should be open. North is up in the air on which way to go. How bout if you are way in the South/southwest of unit, Rawlins might be closest?
Got gas once 6 years ago in Jeffrey City (interesting experience), but that was closed down the last 2 times I went through there. No gas in Atlantic City last time I went through, but there is a place to grab some decent grub.