Let me recall a standard construction.
Up to homotopy equivalence, any map $f: X \to Y$ is a fibering. Take the special case where $X=E$ the total space of a fiber bundle, and $Y$=B, the basespace of a fiber bundle and $f$ is the projection map.
The associated fibration is given by the projection $f'$ from $P_{X \to M_f}$ to $M_f$. Here $P_{X \to M_f}$ is the set of unbasepointed paths from $X$ to the mapping cylinder $M_f$ attaching $E \times [0,1]$ to $B$ at $E\times {1}$.
Now if there is $\gamma$ in the fiber over a point $b$ in the mapping cylinder, which we can choose to be in $Y=B$, i.e. $P_{E \times 0 \to b \times 0}$, then the obstruction to local triviality is the image of the loop $\gamma$ not being contained in $(f^{-1} U_\alpha )\times 0$.
Thus it is not clear to me that this construction, yields a fiber bundle.
When is it true that a fiber bundle can be turned into a fibration that is a fiber bundle?
The map $E \to B$ that is turned into a fibration $E' \to B$ can be turned into a fibration in such a way that $E \to E'$ is a deformation retract. Thus the local triviality condition will still be satisfied on $E'$. See Mosher and Tangora page 84.