From Lee (in hilly East Tennessee):First step (in my opinion) - get lug tires for the tractor. I've had them from the beginning (20 plus years) and I've never been able to detect any harm to my lawn because of them. And they're great (indispensable) on the hills.
Second step - If your tractor doesn't have the brake upgrade, by all means get it. Before I learned of the available kit I had put a drum brake on mine because the 'factory' brake just wasn't good enough.
With those two modifications you have a great 'hill' machine. My twenty plus year old E-15 will still pull 1/3 rick of green oak firewood up a 15% grade hill - in second gear yet!
Is the problem traction on slippery grass? I find that using the weight box in the rear, with a few extra old batteries in it for mass, helps when the grass is wet and I have traction problems. Some people have also reported that traction is improved by replacing the hardened old tires. (If you have nice grass, you probably don't want to go to chains.)