FL600 cruise
M2Dude has it all correct here. It only really happened regularly on the LHR-BGI route (low temps at crz alt, and light weight as you've burnt most of the fuel).
In the MAX Cruise mode the aircraft would sniff about vertically, gently climbing and descending to hold M2 (following the temperature changes), but with an overall general climb.
Heading to BGI you were climbing into very cold air, commonly down to less than -70degs, I saw minus 80 on multiple occasions. (Which was all very good news from a fuel point of view). So the climb was more definite and the likelihood of a subsequent descent lower.
It was pretty easy to spot when the aircraft was definitely going to 'stick' at FL600, and to select Alt Hold, with the already armed AT (with Max Crz mode) engagd in Mach Hold. Very occasionally one may have to subsequently make a subsequent return to Max Crz mode if warmer air was encountered, but I don't recall doing this more than once.
If you had very cold air and a lighter than average load, one would find that the throttles (no 'thrust levers' on Concorde!) were quite a long way back at TOD, maybe 10-14 degrees throttle angle. It was worth noticing this early - the decel/descent was initiated by reducing the throttles to 18degrees, normally this would be from full forward so it can be seen that reducing them by 4 or 5 degrees rather than the usual 18 could result in a disappointing rate of decel. At nearly 20NM a minute this could really screw your descent plan! (I think the record-holders went past BGI still just the wrong side of M1......)
I used to enjoy the last few minutes of the flight at FL600 - before the space station was manned and post-MIR there was a fair chance that at that moment we were the human beings furthest from the surface of the planet. In shirtsleeves, with a cup of tea (or something more palatable if one was in one of the comfy seats).