I'm not aware of cases were Amtrak, of its own free will, ran backup engines. There are certainly cases where the host railroad insists upon it.
When the Pioneer and Desert Wind were still running, Union Pacific used to have a rule that all trains must have two engines in winter, and the Pioneer ran with 2 F40s pulling 3 or 4 Superliners when it had to (and dropped the extra unit the instant UP allowed them to in the spring.)
It will vary run-by-run whether the 2nd engine is added for acceleration, or for grades, or for some other reason. My memory of F40s is that a single unit occasionally pulled as many as 9 single-levels, and 6ish Superliners.
The combined CZ/Wind/Pioneer was limited by grades in Colorado, and started carrying a 3rd F40 when it started carrying a 2nd dining car. This was, far as I know, the only train Amtrak ever ran that routinely required 3 units. In the 1990ish timeframe, the summer consist looked like bag-bag-transdorm-CZ sleeper sleeper coach coach diner lounge; Pioneer coach, sleeper; Desert Wind coach sleeper, diner. Running that 2nd diner through to Chicago started between 1988 and 1990. When the Pioneer started going through Wyoming, there was no longer a need for the 3rd engine between Denver and Salt Lake - I can't help wondering if that was a factor in Amtrak's willingness to split in Denver. It wasn't to improve CZ timekeeping by switching it twice, and wasn't because of Idaho's or Wyoming's enthusiastic political support...heh.
CZ always ran with 2 engines west of Salt Lake, even when it was just bag-trans-sleeper-coach-coach-diner-lounge, while the Builder ran with 1 west of Spokane, same consist minus the lounge.