Check out the following thread with attention to the posts by caterham1700 regarding a DIY Ikea Lak Rack.
Without all the fuss (also the way the legs are constructed has changed over the years making the build much more complicated) I suspect that a stock Lak coffee table may work well with your Rega.
You could also just try the Lak side table if you won't use the extra shelf space (of the coffee table) for other gear.
Drilling the legs and installing adjustable spikes should add some benefit, but you would have to reinforce the holes with 2-part wood filler or something similar (I'd just try it stock to begin with).
Caterhan (Ken Lions) designed and manufactured the custom shelves I use for my TT/CD decks.
Ken always supported the use of light and rigid support for unsuspended decks like the your Rega.
Others are correct in that a better phono preamp should sound, well better, but I prefer to get the most out of what I'm using before moving on.
DeKay