Personally I wouldn't suggest focusing on getting volunteer experience first. Sooner or later, you have to find a paying job, and the faster you start applying for them, the faster you'll get one--really. However, it might be easier getting a volunteer position than a paying job, so there's nothing wrong with applying for both in parallel. Yes, the volunteer job will give you a bit of experience, pad your resume, and help you build your network. So if you end up doing your volunteer job part of the day and spending the rest of your time applying for paying jobs, nothing wrong with it. However, I would strongly caution you against finding a full-time volunteer position that hinders your paying-job application process. I recently had a friend who was working a lot of unpaid internships trying to get experience--he finally had to quit all the volunteer work because it just didn't leave him enough time to look for paying jobs.
Do you have a LinkedIn account? I personally hate social media and resisted for many years. But recently, I forced myself to create a LinkedIn profile. If you fill it out and list your skills and hit the "looking for jobs" toggle so people know you're available for hire, you might even get scouted. Could be an easy way to score a few interviews at least. Be a little wary of the recruiters who contact you, but you should know that they make some money if you end up getting hired by a company they connected you to. So the legitimate ones really want to help you. I'll add that in lieu of job experience, the larger projects that you did as part of your degree are worth mentioning--certainly on your LinkedIn profile as well as your resume.
Do you have a sense of the career trajectory that you want? That helps immensely in terms of building a coherent LinkedIn profile and speaking fluently in interviews, among other things.