In fact, that is what I normally do, but at times I get a bit lazy and try to turn it off with one hand.
Lazy moments are for CD's! There are no shortcuts to good analog reproduction. In fact, the more capable the system, the more care the user must take. My vinyl rig retails for about $18K but I still have to pay attention when I turn the motor on. If I move it by mistake it can take 30 minutes of fussing to re-position it correctly (I use a non-elastic drive belt, so there's zero margin for error).
However, putting something like a rubber mat or tape underneath may allow turning the motor off with one hand, and even further isolate vibrations coming from the motor.
Depending on the system, it may also slur transients and dull dynamic response. The more revealing and accurate the rig, the less tolerant it will be of elasticity *anywhere* in its mechanical connections. Putting rubber feet or even a single thickness of Scotch tape beneath my TT motor turns a vibrant and hugely dynamic table into a sludge. By all means try it, but keep an open mind about the effects you hear.
P.S. Your new cart sounds like a huge step up. (I'm not surprised, most Grados are better at smoothing things over than reproducing lifelike music). Enjoy the journey!