I’d check Facebook and Zillow. Pet friendly is certainly tougher to find and everything gets snatched up right away. Start with apartment complexes and get on waitlists. Follow any instructions to the T and complete anything as quickly as possible. Sometimes you can find off season winter seasonal rentals, especially on the seacoast. Other than that, you’re up against a lot of other people looking for the exact same thing. Some landlords rent to the first person with the money, others have application processes and background checks. There simply isn’t enough housing to meet the demand in southern NH. Good luck to you.