What you are talking about is called workforce housing. A lot of new developments in Providence do have that. It's for those who make 120% of the median income or less (around $83k annually)

The New Paragon Mill development has that for example and they start at $1350 for a one bedroom.

The new building by trader Joe's has them as well. $1600 for a 1 bedroom if you qualify.

Westminster lofts has some workforce units. They had a 1000sq foot unit posted recently for $1710 right downtown.

This building as well https://www.uptonpvd.com/

It's worth emailing any of the new buildings to see if they have workforce housing. I emailed emblem125 (new building downtown) to check and they are offering them too.