Hi!
I am kind of in the same boat as OP (likely Psych PhD reject), but I am unusually pessimistic about my chances with anything so I planned to apply to both MS/MA and PhD programs from the start.
My current mentor suggested this strategy because (at least in my specific field/interest) there are partially-funded MS programs where I could do hands-on research with people who work with the people I want to work with. So, I would do my MS with a mentor who frequently collaborates with people who I would like to do my PhD with. Then, when I've completed my MS, I can hopefully work the connection and show that I can succeed at a graduate level. Also, look at a program's lab websites for any former MS/MA student listings and see where they are now. If they're where you want to be that's a good sign!
Searching for MS/MA in Psych online can be really frustrating, so I would suggest just asking knowledgeable people for suggestions. That's what I did.