Summary: The type III Hermite \(X_m\) exceptional orthogonal polynomial family is generalized to a double-indexed one \(X_{m_{1},m_{2}}\) (with \(m_1\) even and \(m_2\) odd such that \(m_2 > m_1\)) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order \(m_2 - m_1 + 1\), which may alternatively be interpreted in terms of a special type of (\(m_2 - m_1 + 2\))th-order shape invariance property.