So I opted to break rule my own rule and make connections. There were flights through places like Dallas, but they seemed to take longer. Denver was more or less on the way and offered some good arrival time choices. If you do have to make connections, it's good to have at least an hour and half layover in case the first plane is late. Besides, DIA is very new and modern and should be a good place to change planes.
Except, of course, it's in the Rockies. Sure it's not even Halloween yet, but there's not a month in the year it hasn't snowed in Denver, July and August included. It is, in fact, snowing there now.
Rule #2: Never fly through DIA between October and April.