We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to B$ with fiber $F$ where $E \times_G F = (E \times F)/G$ denotes the orbit space and the $G$-action on $E \times F$ is defined ``diagonally'' as $(e,f)g = (eg,fg)$. By doing so we are effectively changing the fiber from $G$ to $F$ for the base space $B$. This is because the local trivialization for the induced fiber bundle is $\psi: p^{-1}(U) \to U \times (G \times_G F)$ (please check whether this is correct) and $G \times_G F$ is naturally identified with $F$.

My question is, could this fiber changing trick be generalized to other fiber bundles? Say for example you are given a non-principle bundle with a nasty fiber $F$ and you want to change the fiber to a nicer one $F'$?

Thank you very much for your attention and help!