I am a little confused. By an external power supply, do you mean an audiophile quality power regenerator? Let’s assume you do mean a regenerator like a Furman, PS Audio, Etc. A power regenerator is designed to produce pure sine wave AC without the electrical and RF noise contaminating a mains supply and house electrical distribution lines. A power conditioner is designed to filter electrical and RF noise from the mains and house distribution lines. Putting a power conditioner after a power regenerator is redundant since a regenerator is theoretically producing pure power. It will not physically harm equipment but theoretically not give improvement and possibly degrade SQ if there interference between filtering stages in the regenerator and conditioner. I can only say experiment and determine yourself if there is an improvement. I only use a conditioner that offers significant improvement in SQ.
However, if you use power supply to mean your mains and house distribution lines, then improvements using dedicated breaker panels, panel surge protection, dedicated lines, and/or better outlets before a conditioner will improve SQ because you are providing cleaner power that will challenge the conditioner less. Output after the conditioner will be cleaner.