This paper extends our previous shape-parameter-free RBF-FD method [1] to the time-fractional Merton jump-diffusion model for pricing European put options. We retain the same spatial discretization: polyharmonic splines of the form r7 combined with complete polynomials up to degree 7 on local stencils of 101 nodes. Weights are computed once through a small augmented linear system. The Caputo fractional derivative (order α ∈ (0,1]) is discretized using the standard L1 scheme, while the jump integral is treated explicitly. To enhance accuracy near the strike price without much additional cost, we introduce a simple residual-based adaptive refinement: every ten time steps, nodes with high residual receive four additional Halton points nearby. Numerical tests on one-dimensional European puts show solid accuracy RMS errors usually between 10−5 and 10−8 for different α with clear convergence as the number of nodes increases. Compared to the non-fractional case and standard multiquadric RBF-FD (which needs shape-parameter tuning), our method is efficient and robust. It is easy to implement and extends naturally to higher dimensions.