Fridge temp for cold bulk / preferment?

So I recently got Peter Reinhart’s Whole Grains book and I am absolutely loving it. Today, I’m trying out the transitional rye meteil, which I’m stoked about since it’s my first time working with rye in an actual loaf. However, one thing I’ve noticed with the cold-rise preferments (biga and soaker) is that they tend to horribly overferment in my fridge. Granted, I leave them in there for 3 days, the maximum amount of time PR’s suggests, and my fridge is really bad and keeping a consistent temperature so that might just be inevitable. Still, I noticed how far they’d gone this morning and instead of bringing them to room temperature before mixing the dough, I just decided to use them straight out of the fridge for fear that they might completely collapse otherwise. Bad idea. The soaker and biga were so stiff from the cold that it took 11 minutes to incorporate, and I can’t imagine all of that mixing is terribly good for a rye loaf, especially a whole grain one. 

I don’t want to repeat this experience but I still want as much flavor as possible to develop in the preferments without them overfermenting. I tried my best to keep the fridge at around 38 degrees, but that might have been too high. Is there a good fridge temperature for a long, cold rise that maximizes enzyme activity and flavor without risk of overfermentation? Or should I simply not leave the preferments in the fridge for so long (<3 days)?