Depending on how flexible you are and can be, there are many opportunities for folks that are during "off hours" especially if the research is more community-based. We do some of our interviews on weekends and late evenings (if its a home visit, and not done at the child's school), and a few of my RAs only come in during the early evening/weekends to do data-entry.
The only roadblock might be lab meetings which are typically held on a weekday between 9am - 5pm, and depending on the PI, they might want you to be there in person, particularly when you are first starting. I also find that you will have a much richer research experience if you work during the times where you can interface with other RAs, grad students, postdocs, PIs, etc. You might have to be flexible and leave work for 1-2 hours per week to join the lab meeting. Is that a possibility, for example, can you make that your lunch hour? A requirement of being in our lab is that you must attend our weekly lab meeting in person, although we are willing to waive this once an RA has established themselves in the lab (for example, if they take on a summer position, we might not require them to attend the meeting every week and instead do a check in during the summer).
In sum, I think if you are open to what experience you can get, just ask around and see what opportunities there may be. I wouldn't drop down to part-time work to get research experience in someone's lab. If anything, try to find paid experiences (e.g., project coordinator), they are out there although they are very competitive, and many times you might be competing with Master's level folks for the position.
Quite honestly I am in a similar situation, and I was lucky to get hired as an RA full-time but before I heard back, I was considering working one part-time job during regular business hours, one part-time job working weekends and evenings, and then volunteer in a lab to get whatever experience I could during the remaining business hours I had free. Would applying for full-time jobs as an RA or lab manager be an option for you?
I feel like, if you are going to do research, the most important thing is to actually be in a lab doing work that you care about and want to do in the future. For myself, being in a similar situation, I wouldn't want to get experience just for the sake of experience if it's not even remotely related to what I am passionate about and interested in.
Also, I second what @entitlement said. I did not have nearly enough psych experience in undergrad so all of the lab manager and most of the RA positions I was applying for, I was turned down because I 1) had only been in a psych lab for 6 months and 2) I was competing with Masters students who had pubs, given talks at national conferences, etc. I knew this and applied to lab coordinator positions anyways because I figured it would be worth it on the off chance that for some reason, someone wanted to hire me, but it seems like your and my previous psych research experiences are pretty similar so an RA job would be the best bet.
I worked full time during undergrad and after graduation (before applying this last cycle). It's definitely tough but doable! I contacted my UG professors (one of which I didn't get the chance to work with due to some serious health concerns) and asked if I could talk about joining the lab. In my experience, the professors were very understanding with regards to working & striving to gain the necessary experience needed to gain acceptance to a solid program. I did a lot of duties from home (screening and scheduling participants, working on posters). Communication is key- let them know up front where you are at, what you would like to accomplish, and how you hope to contribute!
This is a biased opinion, but working your butt off AND dedicating free time to getting yourself ready for a program prepares you well! You'll gain time management skills & learn to balance!
I've known of a number of labs who have looked for someone just like you: meaning someone who can work the off-hours. For example, they run subjects on weekends. You will get less exposure to other staff members, but it is a way to get involved with a busy schedule.