We drove to Banff and met a friend who drove our car home for us -- we were oh so lucky to have such a nice friend. But for most people that's not viable. There is an hourly shuttle bus from Calgary airport to the town of Banff. You could fly to Calgary, or get a one-way car rental to Calgary, drop the car and then take the bus.
The hotels in Banff are quite expensive, >$100. The NPS campground near the town of Banff (Tunnel Mtn campground) is super expensive ($40) and was the worst campsite on our entire trip; it was very crowded and very noisy. There is a great rough campsite about 5 miles south of town, on the route. We didn't note the exact location and may be off a bit in guessing how far south of town it is. It is pretty obvious as it is right next to the river with a picnic table and a well worn flat area for tents.
At the southern end, there is NOTHING around Antelope Wells.
Hachita is the last town before Antelope Wells (45 miles), and Hachita has ~20 homes and no longer has any services other than a post office. There used to be a saloon, liquor store, cafe, and small grocery store, but those are all shut down now. The good news is that there is a very friendly retired elderly fellow - Sam Hughes (POBox91, Hachita NM 88040. 505/436-2662)- in one of those 20 houses. He is a CDT "Trail Guide" and provides a lawn to sleep on, a source of water, and endless stories. He hosts lots of CDT hikers and the occasional GDR rider. He will provide shuttle services as well, for a fee. We didn't talk to him about the details of his shuttle services because we didn't need to use them. You could call him in advance and make arrangements for a ride from either Hachita or the border to Deming (where we believe there is Greyhound service) or El Paso or wherever. Sam told us that some people camp in his yard for two consecutive nights and ride to the border and back in one day, without carrying their gear.
If you don't reach him, you could always call the post office at Hachita to ask about locating him or alternate shuttle drivers. As Sam told us, in Hachita everybody knows everybody; the postmaster lives across the street from him. If you google Sam Hughes Hachita you'll find many references to him from the CDT hiker crowd.
Another potential option is to take a commercial van shuttle service between Antelope Wells and Phoenix or Tucson. We did not call the shuttle companies for more information, although we did see a couple of their vans go by while we were riding between Hachita and the border. The passenger shuttle vans we saw were pulling enclosed utility trailers, so we don't think they would have any trouble taking a bike. The best summary of information about these vans that we found is on the "Traveling to the Village" page of the www.mataortizcalendar.com website. You aren't traveling to Mata Ortiz' village, but some of the shuttles cross the border at Antelope Wells, and you may be able to get on the van there.
We decided to ride our bikes from Antelope Wells to the Tucson airport, pick up a one-way rental, and drive home. It worked out very well for us. The route we took to Tucson was about 300 miles and was roughly half dirt and half paved. We crossed three mountain ranges: one of them (the Chiracahuas) was Amy's favorite place on the entire trip. Most of the ride was on nice quiet roads. The final 10 miles into the Tucson airport was urban sprawl, but even that had a good shoulder. Our rental car (from National) was ~$125 per day taxes included with unlimited miles. We kept the car for two days for the 13 hour drive back to Palo Alto. We liked the fact that we didn't have to find boxes to ship our bikes, didn't run any risk that the airlines or shipping company would damage something, and didn't have to worry about missing bus or plane connections. It was cheaper to pay $250 + gas costs for the car than to buy two plane tickets and pay to ship the bikes or check them in. We also liked the fact that we didn't have any hard deadlines since we didn't have to purchase airline tickets. We reserved our car via Kayak.com from the library in Tombstone AZ, just two days before we arrived in Tucson.
If you reserve a car from National, here's a little advice. The fellow at the counter said that essentially all the vehicles available for one-way rental anywhere in the National system are mini-vans or SUVs. Reserve the smallest cheapest car listed, because you get charged for what you car reserve, not for what car you receive. It probably doesn't matter what you reserve, because you'll get whatever they have available and the one-way rental vehicles come out of their own vehicle pool which is made up primarily of SUVs and mini-vans.