A self-avoiding walk (SAW) is a path that does not self-intersect, and the study of its properties leads to important applications in chemistry, biology and computer networks. We consider SAWs on a restricted square lattice with a finite height equal to 3 and infinite length. We obtain close lower and upper bounds for the number of SAWs of length n and for the connective constant. Additionally, we present a transformation of SAWs on the square lattice to a special kind of walks on the honeycomb lattice. By using H. Duminil-Copin and S. Smirnov’s results for SAWs on the honeycomb lattice we present ways by which close bounds for the connective constant of the non-restricted square lattice could eventually be obtained without the need of thousands of hours of computer calculations.