What I can't figure out is the noticeable hum when I turn on the stage with nothing playing
I agree with @ghdprentice. The phono pre is very important and one at 3 times the cost of the present one will be a big upgrade IMO. However, if you have a ground loop problem which I suspect is the problem, the problem will still exist with a different phono pre.
As usual, @millercarbon has given good advice on how to go about tracking and eliminating the problem. This should be your 1st priority. Then the new phono stage
Then there is the hiss you say is present. If you have the gain set too high and have high sensitivity speakers, you are more likely to have hiss. This is a good reason to list your system so that others can help troubleshoot the problems.