How can I get research experience in between degrees?
As someone who hires research assistants in my own department, here are some concerns I would have about such an offer.


There are students in my own university who would appreciate the research assistantship. My department has more qualified students to do research than supervisory capacity to give them all that opportunity.
I have an aversion to taking on a student who has not made any real commitment to the job and has no accountability. (I will only take on students who are either getting paid or getting credit for their work with me, partly for this reason.)
I strongly prefer to work with students who will stick around for a while, e.g. undergrads who still have a couple of years left, or M.S. students who may be interested in doing a PhD in my department. Working with research assistants initially takes a lot more time and effort than they save me, and I'd rather do it with someone who has a chance of sticking around long enough to give me some return on that investment.


So for me to consider taking on a research assistant in your situation, you'd have to assuage some of those concerns. 


Tell me what skills you already have that will help me and my students advance our current research efforts in the short term. (This obviously requires that you do your homework and know what my group has been working on recently.) 

For example, when you say "I am interested in learning about X, which your students are currently doing," I hear "This is someone who I or my current students will need to spend a lot of time training to do X - which we don't really need, because we already have students capable of doing X - and who may or may not stick around long enough for it to pay off." 

On the other hand, when you say, "I already know a fair bit about Y, and I think it could advance your research in Z that your students are currently doing," I hear "This is someone who offers a promising risk - benefit ratio. We won't have to spend a lot of time on training, and he/she will bring something useful to the table immediately."
Are your plans to go to graduate school fairly firm? Do you have a timeline in mind for applying? If I think that you are committed to the work because you really need it to bolster your graduate school applications, I'm a little less concerned about you disappearing as soon as something gets difficult, or your day job interferes, etc.
Are you seriously considering applying to grad school at my institution? If you are, you should definitely say so.