One thing I noticed a while back is that there were very few organizations serving immigrant populations that rated well on CharityNavigator (at least in Texas). I wonder if you have any insight into why that is, or how to vet service providers in that arena.
Lord, I wish I did. I have had a whole series of (to me) surprisingly bad experiences in that field. In fact I just quit out of what I think is going to be the last one today :-(
I think there is a lot of waste and amateurism in this field -- more than in other fields? I don't know! -- There just really doesn't seem to be a lot of accountability. That is one person's very subjective impression and I don't want to diss these people any more than they deserve.
As a volunteer, two places in TX that I thought had it pretty together were the Val Verde Border Humanitarian Coalition in Del Rio (very fast paced and pragmatic, their main job is to get people fed and onto a bus for San Antonio where they can get more help) (they're probably one of the more conservative organizations, they work closely with the TX National Guard, it's a very mixed staffing situation, which I really liked) ... and the Sacred Heart church's shelter in El Paso ("Casa del Sagrado Corazon"). Long-term housing was too hard to find in Del Rio, so I couldn't stay long, and later I got this bad ankle and it kicked in at the El Paso place (and they couldn't find me anything sit-down to do). It was frustrating to deal with the volunteer coordinators at the El Paso place (I think they went through 3 in <1 year). But they treat the clients great and seemed well organized from what I could see.
Or are you looking more at services for longer-term immigrant populations that are already ~settled in, services like case work and language lessons? That I don't know much about. I don't know why not, I just never drifted in that direction and the couple of orgs I did have an experience at, I wouldn't recommend to anyone.
There's so damn much hype in this field, and the media misses so @#$% much, that I don't know where you could go other than sites like charitynavigator etc. Keep in mind those sites don't always have data on smaller and newer organizations.
The only thing I can think of that you might try is find one that gets a reasonably decent grade for financial stuff and then go volunteer there for a few days and see what you think of it. That would be a lot of effort, of course.
For giving $$ when I have any extra, my go-to is Doctors Without Borders. They got good marks when I checked (a long time ago, should check again) and I don't think they've ever had a scandal.
I hope that's not too TMI and that some of it is relevant to what you're looking for. It's just what I happen to have stumbled across over the years.