El Paso is the Hidden Gem of the World, especially if on a bicycle. I am an expert bicyclist, and pedal virtually everyday while here (please follow me on Twitter). I've been working with the authorities, particularly the MPO, in establishing bike routes, and am currently pushing for an effective East-West transit route between Ft. HancockTX and Las CrucesNM. As of right now, there it is a challenge to get through, whether going across country or across town. El Paso does not yet have any designated route and there is ongoing road construction which makes getting through a challenge.
However, this is what I suggest: Do not pedal I-10 between Ft. HancockTX & Las CrucesNM. If Westbound, depart the good shoulder (it is a DISASTROUS shoulder a bit further West) at Ft. Hancock/Spur 148. If you've pedaled from Van Horn/Sierra Blanca, you'll want to eat at Angies Restaurant (beside the Shell station), cash only, and possibly overnight in the hotel. Go South on Spur 148 downhill to intersect with TX20/Alameda. If you have your passport, turn East and go to the POE, cross into Mexico, then visit el Porvenir or go West on Ruta Dos. Otherwise, go West on TX20/Alameda to Fabens/Farm Road 258; go Southwest to San Elizario/Main Street/Historic Zone; go West on Glorietta Road to FR258/Mission Trail (this is often congested, but better than either TX20/Alameda or North Loop/76) to Schutz Road to Padilla Road to Mecca/Independence Road to Elvin to Lee Starling to Independence to North on Yarbrough; turn West on North Loop to Delta Drive to US62-85/Paisano/CanAm Highway to Racetrack Drive exit; continue North/West on Doniphan Drive/TX20/NM478/Main Street; turn West on NM373/Union Avenue to NM28/Mesilla; turn North to NM3359/Calle de Norte; turn West and cross the Rio Grande, where NM359 becomes Fairacres Road going North; pass underneath I-10, go West on Picacho Avenue/US70, ascend out of the valley, join I-10 West (shoulder) going toward Deming.
If you are pedaling West to East, reverse the above directions.
This route is relatively flat although it can be really bad & congested when in El Paso.
You can take NM404/Ohara Road over the gap into ChaparralNM/NE El Paso/access to US54, but that, too, can be congested & challenging, especially if you go into El Paso. If you choose to go that way, continue South on NM213/FM3255/MLK drive to Kenworthy to Loop 375/Purple Heart. There is current construction at Liberty Expressway/Spur 601/Loop375/Purple Heart, bicycles are banned South of this point, so you have to turn West on Spur 601. The nest closest road to continue South is Global Reach, a connector between Montana Avenue/US62-180 and is 3 miles/5kms of terror during rush hour. You can only avoid it by not pedaling it during rush hour; for me, I've had to be on it NLT (05:00), M-F, because you don't want to get in the way of a soldier motoring crazily trying to make muster!
If you have a route that puts you on NM28 South of Old Mesilla to the TX state line, AVOID IT; this may be a flat road, but it is EXTREMELY HAZARDOUS=narrow 2-lane farm road in an increasingly more developed suburban area!
If you have a route that puts you on TX20 between I-10 and Alameda, AVOID IT; this road may be 3-lanes divided, but it is Global Reach exponential and is hilly! For years, I've been pushing the MPO to at least put "Share the Road" signs on it, which is just about the very least one amount of safety possible.
Until road construction is completed, and that meets the minimum guidelines for multi-modal use, stay alongside the river. It will be a while, but it is possible to get through El Paso with a certain amount of discomfort.
I've lived here for 25 years now, have pedaled every road into/out of/international & local, and am quite knowledgeable about the area. Feel free to contact me if you have an questions.