This is motivated by the proof of Theorem 6.8 in Bott and Tu's Differential forms in algebraic topology. In the second paragraph there, they make a claim which can be generalized as follows:
Suppose we have two topological spaces $Y$ and $I$. Suppose we have a section $s$ of some vector bundle $E \rightarrow Y \times I$ (which is the bundle of linear maps between two bundles in the book), which is defined only over $Y \times \{ t_0 \}$, with $t_0 \in I$ fixed. The claim is that, if $Y$ is compact, then $s$ can be extended to a neighborhood of $Y$ in $Y \times I$.
The argument is that we can find some open sets $U_1, \ldots, U_n$ in $Y \times I$ over which $E$ is trivial and such that $Y \times \{ t_0 \} \subset \bigcup\limits_{i=1}^n U_i$. Obviously we can extend the section $s$ to sections $s_i$ over each $U_i$ using the trivializations. But how can we ensure that these $s_i$ glue together to a section on the whole $\bigcup\limits_{i=1}^n U_i$?