Power Supply Issues in Cold Temperatures
Troubleshooting something like that. The change to a few volts is more than I would expect even if the e-caps were completely open. 

Inspect visually for a bad solder joint or cracked component. Pay particular attention to the control circuitry unless you believe the ripple voltage at the output is increasing. Look also for evidence of heating in other areas and evidence of water damage if that could be a possibility. 

If you don't have a schematic, make one. It doesn't have to be perfect at this point, just identify the blocks and where they interface to each other. 

Instrument the power supply (eg. bring out the feedback voltage divider voltage) and either put it in an environmental chamber and cool it, or leave it outside with a device to measure and log the voltage. Examine the logs for clues. 



If the power supply can be replaced by a switching power supply (does not have to be especially low noise, nor does it have to have high current surge capability), consider just buying an open-frame supply from a reputable maker. 

Motors typically require large surge current at start-up and if you try to replace an old transformer-rectifier linear regulated supply with a switching supply  you may need to use one that is significantly higher in current capability than the nominal rating of your existing supply.