We study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multi-sections. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold M, under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho-Joyce, which concerning non-Hamiltonian invariant property of the immersed Floer cohomology. Furthermore, if M admits a Lagrangian torus fibration over an integral affine manifold, we prove that, under some additional assumptions, this new equivalence is mirror to isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform.