Installed hundreds during the '70s and '80s. Some were difficult, others were easy-peasy. If the starting 'tail' is just too thick, you could try scraping(burnishing?) the backside so the overlap is not so pronounced. Pulling like crazy will stretch the part of the grip that needs to conform to the larger circumference. You can't completely eliminate the 'bump' that results from the overlap; but it can be minimized by these techniques. Plenty of use held/hold the racket one way to use or avoid the bump feel-wise.
Probably installed 98% of the leather grips without any adhesive. Staple in the butt-cap, pull tight while wrapping, staple at the top and Bob's your uncle! I'll admit that some well-placed carpet tape came in handy on plastic handles like the Wilson T-series and Head Competition models.
Careful installing too many leather grips if you have tennis elbow issues. It's a world of difference from synthetic grips in the amount of pull required.