Most guys will tell you to run some type of ag/bar lug tire. I have a Sears Subruban that does everything from mowing duties to plowing a garden to grading with a rear box blade and moving material with a front dump bucket. I have 23x10.50 Deestones and they are a decent all purpose tire. You may hear the ags will tear up your yard and yes, if you start spinning they will dig, but in my opinion they hold traction alot longer than a pair of turfs like you have. When turfs spin, they tear up some grass in a hurry. I am much happier and have less damage when using my ags. Wheel weights and adding fluid inside tires helps with traction also.
In your case, I would also consider some Carlisle Field trax, but they are $$$$$$
Here are some possible tires for you.
These are ag tires, but the lugs are a little shorter and rounder shoulders. We use on our church Zero turn and they grip, but less damage when spinning around those tress.