Functional bootstrapping seamlessly integrates the benefits of homomorphic computation using a look-up table and the noise reduction capabilities of bootstrapping. Its wide-ranging applications in privacy-preserving protocols underscore its broad impacts and significance. In this work, our objective is to craft more efficient and less restricted functional bootstrapping methods for general functions within a polynomial modulus. We introduce a series of novel techniques, proving that functional bootstrapping for general functions can be essentially as efficient as regular FHEW/TFHE bootstrapping. Our new algorithms operate within the realm of prime-power and odd composite cyclotomic rings, offering versatility without any additional requirements on input noise and message space beyond correct decryption.