This work extends Favard-type spectral representations for banded matrices \( T \) beyond the bounded setting. It assumes that, for every \( N\in\mathbb N_0 \), there exists a shift \( s_N\ge 0 \) such that the shifted truncation \( 𝐴_𝑁 := 𝑇^{[𝑁]} + 𝑠_𝑁 𝐼_{𝑁 +1} \) admits a positive bidiagonal factorization (PBF). Allowing \( s_N \) to depend on \( N \) leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with \( A_N \) are translated by \( x\mapsto x-s_N \), producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal \( J \) with positive sub- and superdiagonals, each truncation \( J^{[N]} \) admits a shift \( s_N\ge 0 \) such that \( J^{[ 𝑁 ]} + 𝑠_𝑁 𝐼_{𝑁 +1} \) is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for \( J \).