Before encrypting a letter the first rotor advances by one, right? So there could be a way, once the first rotor turns 26 times, make the second rotor advance two positions instead of one. Or three. Did the Enigma take advantage of it?
No, there was no way on the Enigma machine to change the behavior of the rotor rotation.
Rotation on the Enigma was a fixed mechanical property of the rotors like the gears in a clock. However, other cryptographers noted this vulnerability in the Enigma and improved it in follow on rotor machines particularly the British Typex and the American SIGABA aka ECM Mark II aka M-134.
The Typex had five non-reflecting rotors (watch this video for why reflecting rotors were a flaw in the Enigma), but the first two were stationary. They could be set and acted like the Enigma's plugboard but did not have the flaw of being reciprocal, a key means of defeating the Enigma's plugboard (watch this video for why).
The other three rotated, but Typex rotors came in two pieces. An wiring insert which scrambled the letter, and a sleeve which determined when it rotated. Each sleeve had different numbers of notches in different locations making it difficult to work out when the rotors would rotate. They could even be reversed giving yet more variation.
SIGABA also had varying rotation but electric signals rather than mechanical notches. The original used a paper tape with holes in it to allow or block an electrical signal. Each keypress read and advanced the tape which could contain many holes. Each keypress could turn any and all (or none) of the rotors.
The next version did away with the paper tape and instead used yet more rotors (control and index rotors) to generate a 5 bit signal, each bit indicated which scrambling rotor (code wheels in the illustration below) to turn. This eliminated the paper tape and instead rotation was determined by setting the rotation rotors according to daily code.
This provided SIGABA with very good pseudo-random behavior for the time and it is not known to have been defeated during its service lifetime.