Nothing really to offer other than it was this way last Summer when I was moving as well. The inventory wasn't horrible in the southern part of the state, but further north it was rough. I wanted to rent no further south than Concord and no further north than Berlin so I could set up a base with which to leisurely search the area to buy/build a home.
My after work activities consisted of living on the Zillow, Realtor, and Trulia apps as some update at different speeds. I lucked out renting a great little SFH by Squam lake for slightly more than $2K a month. It checked all my blocks except for cell coverage(0 bars here). The area is a bit ritzy and touristy for me, but the location gave me an ideal base and the homeowner(landlord) is great. I contacted the listing agent directly within an hour of it popping up on realtor.com. Essentially went to her company's website where it had her direct email and shot her a note. That evening I had an application in my email and the next day I had an approved application.
I obviously didn't want to pay that much to rent, but SFH rentals were almost non existent and those that did appear disappeared quickly.
I've moved around all my life as part of my career and have never had a problem finding a suitable place quickly. This was the hardest move by far. Looking back on it, I should have also browsed each major company's website to see their listings before it hit the MLS creeping apps.
My question from last year on rentals with a couple of responses indicating it was difficult then too.
https://www.city-data.com/forum/new-...entals-nh.html