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Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitelymany specializations where its Picard rank jumps, hence extending our previouswork with Shankar--Shankar--Tang to the case where $X$ might have potentiallybad reduction. We prove a similar result for generically ordinarynon-isotrivial families of K3 surfaces over curves over$\overline{\mathbb{F}}_p$ which extends previous work of Maulik--Shankar--Tang.As a consequence, we give a new proof of the ordinary Hecke orbit conjecturefor orthogonal and unitary Shimura varieties.

Picard rank jumps for K3 surfaces with bad reduction