I also use a tub mat. But, after every few uses, the suction cups won't stick because mold develops under them because they never dry completely. The mold is very slippery. When that happens, I take the mat outside & clean the underside with bleach & also clean the tub. It's also a good idea to keep your weight directly over the mat when getting in & out of the tub & not stepping in at an angle. Most tub mats won't work if the tub has a textured surface.
I'm having my tub/shower converted to shower only so the step will be very low. Much safer than having to step over the edge of a tub.