Success with replacement grip application boils down to having an orderly start. Where applying an overgrip is somewhat simplistic and mechanical, substituting a replacement grip is more of an art. I find that if I can get the initial few wraps around the handle flat and secure, the rest of the grip usually takes care of itself.
Most replacement grips come with some paste or adhesive on the starting flap to help secure it in place to begin a smooth spiraling process. However, it’s often not enough grab to withstand a tight pull of the grip—you can use your thumb to fasten it to the handle, but that can get in the way on the first turn.
Instead, I prefer to use a staple gun to anchor the starting flap to the frame. (After all, that’s how manufacturers apply grips to their racquets). Just make sure to purchase one that can shoot 1/4 inch (6 mm) staples—and if you can find a smaller size, all the better.
With the starting end bolted to the handle—the staple should sit as flush against the grip as possible—you can pull the grip taut on the first wrap to avoid any bunching or lack of stability (grip slide). It also promotes a neat and minimal overlap to prevent the bulkiness and rounding of the bevels that can occur with chunky wrapping. Essentially, you’re getting as close as you can to a factory-installed grip.