You could easily convert your current bike to an internally-geared hub for the budget you stipulate. The bike would have a similar aesthetic to your fixie - simple chain line. That's what I'd do.
It is difficult to get much of a bike for $350 in the new market, and if you hunt around you can find a decent bike on the used market.
If by "fixie/single speed" frame, you are referring to rear track forks (axle slot to the rear) the only ones like that that I know have derailleur hangers would be Surly and On One.
If you simply mean "road type frame", then derailleurs (gears) are more easily accommodated. Here's a pretty decent deal for a new bike:
http://www.bikesdirect.com/products/...e_xi_steel.htm
I lived in Vegas for a couple of years and commuted. I lived in Summerlin and worked on the valley (not the strip), with about 800 feet difference, so my ride home involved a pretty steady low-grade incline.
Technically, a 3-speed hub would do the trick and offer the most simplicity.
If you want to stay fixie, Sturmey Archer makes a 3 speed fixed gear hub (
the S3X) that offers sufficient gear range to get you where you need to go, IMO. I've used one and found it to work great. Some freewheels will also fit, so running the hub with a FW is an option too.