Numerical solutions to partial differential equations (PDEs) are pivotal in tackling challenges across various scientific and engineering fields. The selection of solution methods depends heavily on factors such as the type of the equation, domain characteristics, and boundary conditions. For example, finite difference schemes directly solve the PDE formulation, while finite element methods work with their variational counterparts. Often, equivalent integral equations are also solved using techniques such as the Nystrom method. Certain PDE solvers have recently been designed using the Fourier continuation principle (FC). For example, the PDE solver in [1] utilizes an FC construction to resolve the Gibbs phenomenon. Another FC construction based on Hermite polynomials [2] has also been used in the context of approximation of integral operators and Nystrom method. In this paper, we first discuss a Fourier Continuation construction that employs the two-point Hermite polynomial. We then present a modification of the FC-Gram construction outlined in [3] where the SVD-based extensions of the Gram polynomials are replaced by the one that utilizes the two-point Hermite polynomial. While the modified construction achieves similar accuracy levels to that of the FC-Gram, it additionally comes with a rigorous convergence analysis that is not available for the original FC-Gram. We present a variety of numerical experiments to demonstrate the performance of the modified method.