I have degrees in EE and CS. I think it would be very difficult to fall back to an EE job if you've been focused purely on software. If your software career is low level work that is EE adjacent then the EE education can add value.

At my school Computer Engineering required several classes that were optional for EE but I would consider them essential for computer hardware work. Algorithms and data structures, operating systems, and computer architecture (modern out of order/superscalar).

The classes you take matters a lot more than the name of the degree, and good employers know that.