The Wikipedia definition of Secondary vector bundle structure is based on local coordinates. Maybe a more intrinsic definition can be given.
The idea is the same as what Wikipedia does. $(E,p,M)$ is a vector bundle, $p_*$ is the pushforward of $p$, then the fiber-to-be $p_*^{−1}(X)$ at a given tangent vector $X$ in the tangent space $T_xM$ of a given point $x\in M$ is the union of a series of affine spaces $p_*^{−1}(X)|_v$ at each point $v$ of the fiber vector space $E_x$. If these affine spaces can all be identified with the same vector space $V$ then $p_*^{−1}(X)$ can be identified with $E_x\oplus V$.
To identify every affine space $p_*^{−1}(X)\rvert_v$ to the same vector space, use the "connetor map" described in the Wikipeida page. First choose an Ehresmann connection, then first project elements of $p_*^{−1}(X)\rvert_v$ to the vertical subspace $V_vE$ using the given connection, then identify the results to elements in $E_x$ by the inverse of the vertical lift $vl_v$. So the identification destination is $V:=E_x$ and a fiber-to-be $p_*^{−1}(X)$ can be identified with the vector space $E_x\oplus E_x$ so become a vector space.
Cab the above way define something equivalent to the definition in Wikipedia?