Never, ever book a connection to another carrier on the same day as arrival of any North American long distance train, Amtrak or VIA. Especially a critical one like an international flight.
Lay over overnight.
This recommendation applies at all times of year.
If you are going to take one western Amtrak LD, take the California Zephyr, hands down. The scenery is spectacular, the Colorado Rockies, the canyons of the Western Slope of the Rockies and California's Sierra Nevada. Avoid the Texas Eagle, between Chicago and San Antonio, TX the train has poor amenities, no Sightseer Lounge and lousy "Flex" reheated meals. To top it off, the scenery is "meh" at best. The SW Chief is a full service train like the Zephyr and the scenery is good, Raton and Glorietta Passes and red rock country, but it is no match for the Zephyr.