I think that the proposition still holds for smooth finite-dimensional manifolds. Here is a sketch of an argument.

First of all, for a bundle of rank $k$ the following statements are equivalent.

- there exists a complementary bundle of rank $\ell$,
- the classifying map $X\to BGL_k(\Bbbk)=G_k(\Bbbk^\infty)$ factors through $G_k(\Bbbk^{k+\ell})$.

Next, consider an $n$-dimensional smooth manifold, then $M$ admits a triangulation. The stars of its barycentric subdivision provide an open cover $\{U_0,\dots,U_n\}$ of $M$. Here each $U_i$ is the disjoint union of the stars of those vertices that come from the $i$-simplices of the original triangulation. In particular, each $U_i$ is a disjoint union of contractible open subsets of $M$. Fix trivialisations of $E|_{U_i}$ for each $i$.
Then the transition maps $g_{ij}\colon U_i\cap U_j\to GL_k(\Bbbk)$ can be used to construct an explicit classifying map from $M$ to the $n$-fold join $(GL_k(\Bbbk)*\cdots*GL_k(\Bbbk))/GL_k(\Bbbk)$ in Milnor's construction of a classifying space.

Now, there is a homotopy equivalence
$$\operatorname{colim}_n(\underbrace{GL_k(\Bbbk)*\cdots*GL_k(\Bbbk)}_{n+1\text {factors}})/GL_k(\Bbbk)
\to\operatorname{colim}_\ell G_k(\Bbbk^{k+\ell})$$
between these two models for $BGL_k(\Bbbk)$.
Because each of the joins on the left is a finite CW complex, the restriction ends up in a finite-dimensional Grassmannian. In particular, the classifying map $f$ above factors through one of them. Together with the observation above, that proves the claim.