El Paso (Texas)
El Paso is a town in Texas right next to the Mexican Border and at the border to the state of New Mexico. It's a very cheap city with one of the lowest crime rates of any big city in the country. It is generally a fairly easy city to get out of.
West on I-10 and North on I-25
From downtown there is a $3 bus that runs to Las Cruces, New Mexico.
Heading northwest, I took a city bus to the El Paso Outlet Mall. Was able to get a ride 6 miles further out of town, then was able to get a ride to Las Cruces.
East on Highway 10
Take the bus going to Horizon and ask the driver to be dropped of at the intersection Horizon Ave / Highway 10. Around the intersection are three gas stations with lots of trucks going east. Take care to look like a regular guy and the personelle of the gas stations won't bother you. Alternatively you can use a sign next to the on-ramp.