I have children in both Santa Fe and Denver, and have been investigating what was in the good ol' days the Chief Connection.
Greyhound appears to have at least two services between Denver and Raton, both of which appear in the SWC time table. One departs Denver Union Station at 5:45AM and arrives and terminates at Raton at 10:00AM in time to catch #3 at 10:56. The second departs Denver (but not Union Station) at 10:00 and stops at Trinidad (11:40AM and Raton (12:25PM), too late to catch #3, but plenty of time for #4. It continues on to Albuquerque, arriving at 4:20PM lagging behind #3 which arrives at 3:55.
Strangely enough, Den - Raton - CHI is a published route, but involves the early bus requiring a much longer layover.
For the reverse, #4 departs ABQ at 12:10PM and arrives at Raton at 4:50PM. A Greyhound originating there departs at 5:05PM and arrives Denver Union Station at 9:10PM. Another bus departs ABQ at 2:55PM and Raton at 6:50PM and arrives Denver (but not Union Station) at 11:20. During a smoke break at Raton recently, the 5:05 bus was loading passengers from #4 and I asked the driver what happens if it (#4) is late. I believe he said they wait 1 1/2 hours. After that, I suppose the 6:50 is an option.
So if we decide to do the circle tour (originating in GBB), we'll hit Santa Fe then Denver since the 9:10PM arrival in Denver is more attractive than the 5:45 AM departure.
Note that the DEN - RAT bus connecting to #3 and the RAT - DEN bus connecting from #4 does not stop in Trinidad or La Junta, at least according to the Greyhound web schedule.
I found discrepancies between what the Amtrak schedule reads, which was correct in going from Denver to Raton, but the Greyhound schedules show different times by as much as several hours. Very confusing if you try to figure this out looking at it from Greyhound's website. The day I rode it in October, other than arriving at the Denver Union Station about 15 minutes late, the ride to Raton was just fine. I will be using Greyhound in May when I get off the westbound Sunset at Tucson, stay overnight and return to Flagstaff the next day, with a hopefully not messed up connection in Phoenix.