I'm going to assume you have an S-Series since you didn't specify.
If you didn't get an alignment, I'd have it checked. It's amazing how slight differences can affect steering feel.
There is a joint where your steering shaft connects to the steering rack under a clear plastic boot. This is easiest to see or get to with the drivers' side tire and inner wheelwell liner removed, but you can probably get to it laying on your back under the car looking up. There is one bolt that connects that shaft to the rack. If it is loose, there will be play. Make sure that that bolt is at the proper torque spec. Other than this joint, I don't think there is any way to 'tighten' up the gears or anything else for that matter unless the rack is serviceable and you can order internal parts for it. I'd be surprised if that is the case. BTW, if you want to try a different rack, I have two for sale. Details are in the for sale section of my website.
I just tend to think your new tires have sidewalls or construction that may make the ride cushier or quieter, at the expense of a bit of precision or stiffer feel you got used to in the steering. I can feel this when I switch between my summer tires (Toyo T1S) and winter tires (Bridgestone Blizzak). The winters are easier to turn, but the steering doesn't feel as connected or precise.
Lane [ l a n e @ p a i r . c o m ]