In recent years, Heegaard Floer theory has proven an invaluable tool for studying contact manifolds and the Legendrian and transverse knots they contain. After surveying a bit about the connections between transverse knot theory and branched coverings, I will discuss a method for defining a variant of Heegaard Floer theory for infinite cyclic covers of transverse knots in the standard contact 3-sphere. This invariant takes the form of a Z[t,t^-1]- module and generalizes one defined in joint work with Baldwin and Vertesi for transverse knots braided about open book decompositions. In this talk, I will discuss how our invariant is constructed and present some basic properties. This is joint work with Tye Lidman and Sucharit Sarkar.