We show that given an infinite triangulation $K$ of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than $π$ at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of $K$. Moreover, if $K$ is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation $K$ and a conformal class $X$ such that there are infinitely many conical hyperbolic structures in the conformal class $X$ with a circle packing in the combinatorics of $K$. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.