Soft memory foam is cheaper than firm memory foam, so most of the memory foam mattresses and toppers being sold are the soft ones (and they often don't specify their density). The higher the pounds per cubic foot, the firmer the foam. Look for 5 or more lb/cu.ft. density. 6" or so should be plenty IMO. If you get a firm memory foam and it doesn't feel quite plush enough, add just an inch of less dense foam on top of it.
Or, get a Froli system and top it with maybe 4" of memory foam. The Froli springs connect together for variable sizes and configurations, and can be set for the desired firmness. And added benefit will be the way they hold the foam (or whatever mattress material you put on top of them) up from the platform and give you the ventilation you want for mold prevention. Kill 2 birds with 1 stone.
But I recommend that you immediately start some Kegel exercises. You can lie down flat on the floor on your back, then thrust your pelvis to push the small of your lower back down toward the floor... keep breathing, and hold for as long as you feel you can. Let up, relax, wait a minute, and repeat. Do this about 3-4 times each day for 1 week, and see if your back pain lessens or goes away. See, the most common cause of lower back pain onset is the abdominal muscles becoming too weak to hold the back in proper position. The Kegels strengthen those muscles. This might at least tide you over until you decide what change you're going to make to your bed.
My home mattress is a Select Comfort (the sleep number mattress without the digital readout) with 3" of memory foam on top. When camping, I have 2" of 'Tulsafoam' (quite firm stuff, but Superfoam is even firmer) topped with 4" of memory foam.
