I live in Texas, and like to backpack in the Whites. I've tried both Manchester and Boston.
My preference is to get the first non-stop flight out of Houston into Boston. This gets me into Boston around 1 PM. I get a rental car, and drive to Pinkham Notch Visitors Center (PNVC), and spend the night at Joe Dodge Lodge. From PNVC, I can take the AMC shuttle to a number of trailheads. I can also take the shuttle back to PNVC if I don't hike back. The AMC shuttle runs from May to October. The rental car stays at PNVC. I can buy stove fuel at PNVC or EMS in North Conway.
When I'm done backpacking, I drive back to Boston, and get a late afternoon flight back to Houston.
There is a bus to PNVC from Logan airport, but it arrives fairly late at night. I believe the bus also stops in North Conway. There is another bus route that stops in Lincoln, NH.