In part it depends on the kinds of programming you want to apply for. There are three kinds of programs:
1. Simple programs that largely revolve around "business logic" and writing and reading from a database. The most complicated stuff is either modeling the whole damn business process or maybe generating some reports. This stuff CAN become complicated, but I treat that as a design flaw. You will have a hard time getting into this without HR buzzword compliance. I think they're doing you a favor. Lots of poor CS students and business/IT majors end up in this place.
2. Simple programs that have complicated results. Lots of EE work revolves around designing simple circuits or programs and demonstrating they satisfy certain engineering properties. Control loops and whatnot. The programs are simple but the consequences may involve differential equations or recurrence relations. The programs are generally a variation of a small set of algorithms, with parameters tuned for your design's requirements. This is where EEs turned programmers generally end up.
3. Complicated programs. Wicked applications of math, like Hidden Markov modelling, or optimizing compilers, language theory and software verification. Speed reading CS won't count here, as you need to deeply understand the principles of computers and programs before you'll be productive here. If you've written a Fast Fourrier Transform implementation in school, you might have a shot. Otherwise, top CS students and grad students only.
So my advice is to find firms that do place EEs into programmer roles. Embedded systems places mainly. I have many EE friends that landed in programming roles, even though they insisted they would fight the trend. Try to position yourself in your current EE employment to work alongside programmers, learn what they do and pick up a few tasks for them. Eventually your CV will have programming related material to pull from for a targeted resume. And you'll have the friends and contacts to pass your resume directly to hiring managers for consideration.
If you're after category 3 work, consider getting graduate degree (don't waste your time on another bachelors). Plenty of EEs go back for different degrees. Look for institutions with EE and CS in one department, and I personally recommend against financing it. Spend 10s of thousands of someone else's money earning an advanced degree. There's TA positions, research positions, scholarships and employer paid options. Be prepared to sacrifice some portion of your life; be it your family, your career or your sanity.