Could be a lot of things:
Power supply - The 060 draws less power than the 040, but such an expansion is still too much for a vanilla supply, especially iwith a HD installed. Are you using a higher wattage one?
It's an old A500 brick rather than the newer A1200 stock PSU.
Dirt - not saying your machine is filthy, but grime and muck always gets into connectors. Have you removed the card and the RAM, blasted any dust away, cleaned the connectors and put everything back as it was?
I should probably go over it with the air duster at some point, the floppy drive was playing up the other day as well, I blew in it and the disk was no longer unreadable :/
Heat - While the 060 is generally cooler than the 040, some XC models ran pretty hot. Is your system sufficiently ventilated?
The room is fairly cool, there is no radiator in here and it is Scotland in the winter! If it was heat I'd have expected the problem to get worse the longer the machine was switched on, but the opposite is the case. After it's been on for 15 minutes or so it's fine.
Age - drying/leaky capacitors, dull solder joints, flat/leaky batteries. These are all sources of potential instability.
I can take a look... not had the case open in a long time now though, but I was meaning to clean it some day so maybe now's my chance.