Bathtub mats help you keep a firmer footing in the tub or shower. If you have a reglazed tub, don't use plastic or vinyl mats with suction cups. The suction cups could scratch the tub or even remove some of the reglazing. A better choice is a mat with no suction cups, a cushioned vinyl mat, a cork mat, or even a cloth mat or towel you can hang to dry after each use. You want something that is soft enough not to scratch the tub.
You can purchase various decals and tub strips designed to reduce slipping. These may work well in your tub; however, if you try to remove them at some point, they may damage the reglazed surface. So if you plan to use this solution instead of a bath mat, you should consider them a permanent addition to your tub, not something you can change without risking damage to the reglazing.
You should clean your tub with a nonabrasive liquid cleaner. Use a small amount of cleaner, and rinse the tub thoroughly afterwards. Avoid any cleaners containing bleach or ammonia, and don't use harsh chemicals such as dishwasher or laundry detergent in your tub. Avoid any cleaner that leaves a slick residue. Wipe the tub surface dry after cleaning, which will reduce the slipping hazard in the tub and may allow you to use the tub without a bath mat.
Some reglazing manufacturers recommend you wax your tub regularly to maintain the glaze. Waxing helps to repel dirt and maintain shine, but it also makes the tub more slippery. If you're concerned about a child or the elderly or someone with balance problems using the tub, installing grab bars can add extra stability. You might also ask the reglazing company about spraying the bottom of the tub with a coating to make it less slippery.