In this article, we present a method to construct a positivity-preserving numerical scheme for a jump-extended CEV (Constant Elasticity of Variance) process, whose jumps are governed by a spectrally positive $α$-stable process with $α\in (1,2)$. The numerical scheme is obtained by making the diffusion coefficient $x^γ$, where $γ\in (\frac{1}{2},1)$, partially implicit and then finding the appropriate adjustment factor. We show that, for sufficiently small step size, the proposed scheme converges and theoretically achieves a strong convergence rate of at least $\frac{1}{2}\left(\frac{α_-}{2} \wedge \frac{1}α\wedge ρ\right)$, where $ρ\in (\frac{1}{2},1)$ is the Hölder exponent of the jump coefficient $x^ρ$ and the constant $α_- < α$ can be chosen arbitrarily close to $α\in (1,2)$.