Why do my experimental data for Hooke’s law not match the expected data?

The coils of the spring are touching one another and the spring is initially under self-compression so it takes a finite force to move all the coils away from one another and for the spring to behave as you expected.
That initial part of the source vs stretch curve is real (you had obtained data whilst undertaking an experiment) and hence should not be ignored.

There is no "systematic error" in the experimental results. I have done this experiment as well when teaching a high school physics class. The effect you are seeing is due to the fact that the spring you are using requires a small amount of force before it begins to stretch, while the "theoretical" spring that is described by Hooke's Law does not. This effect for your real-world spring is real, and may be due to heat treatment or other handling of the spring material when it was coiled into the spring shape upon manufacture.

Note that you should not assume that the graphed data should be curved downward in order to force the graph through the origin, as this is not what you measured.