We have carpet. Do they make something that can go on top of carpet so that the patient lift can move easier?

I know this works,,, masonite the firm kind smooth side up, thickness depending on the weight of the load
5-16ths will hold a horse in shoes as long as it doesnt get wet, cut around obstacles and duct tape the joints

there are commercial runners that are more firm than carpet in the home.
There are hard plastic runners that might be used.
If it is just 1 room that is in need of change you could get vinyl flooring (the kind that is in large rolls, not the plank style) and cut a piece to fit the room. If that works and you need more cutting a piece that would run down a hall might work.

When my husband was under hospice care in our home, and completely bedridden, I spoke to them about getting a Hoyer lift to help getting him out of bed to put on bedside commode, and they told me because my house is fully carpeted that a Hoyer lift would ruin my carpet, as it sounds you might be finding that out as well. I ended up hiring an aide to come in the mornings to put him on the bedside commode, which worked great as she just used gait belt to lift him. I guess you can also try what NeedHelpWithMom suggested with putting some plywood down underneath it. You may want to ask the folks that supplied your lift to see what they might recommend. Good luck.

I did a Google search. The most common answer was to place a piece of plywood on the floor. One person said that they also cut a piece of linoleum flooring to put on the top of the wood.

One person said their doctor advised against using chair mats. So, I suppose that you could try the plywood.

Best of luck to you.