$P:M\to N$ is said to be a $C^k$ fibered manifold if $M,N$ are $C^k$ manifolds and $P$ is a $C^k$ surjective submersion. A $C^k$ fibered manifold is a $C^s$ fiber bundle ($s \leq k$) if it admits $C^k$ local trivializations: i.e., there exists a $C^s$ manifold $F$ such that for all $x \in N$, there exists an open neighborhood $U$ of $x$ and a $C^k$ diffeomorphism $f: P^{-1}(U)\to U \times F$ with $\pi_1 = P \circ f$, with $\pi_1$ being projection onto the first factor.
Question: Let $P:M\to N$ be a $C^\infty$ fibered manifold which admits $C^1$ local trivializations, so $P:M\to N$ is a $C^1$ fiber bundle. Then is $P:M\to N$ necessarily also a $C^\infty$ fiber bundle? I.e., do there exist $C^\infty$ local trivializations?
Motivation: If $M, N$ are two $C^\infty$ manifolds which are $C^1$ diffeomorphic, then they are $C^\infty$ diffeomorphic (see, e.g., Differential Topology by Hirsch, Chapter 2). However, I am not sure if this result can be extended in the context of fibered manifolds and fiber bundles to ensure that the approximating $C^\infty$ diffeomorphisms preserve fibers.
Note: This question is related to the question I asked here. However, it is different because while I am still assuming here that $P:M \to N$ is a $C^\infty$ fibered manifold, here I am not a priori assuming that $P:M \to N$ has the structure of a $C^\infty$ fiber bundle.