While some people stay into the first week, the majority start to head home on 1/1. Where NYE falls in the week also can have some impact on how quickly folks head home. This year, for example, it's a Tuesday. Many people will need to be to work and school on Thursday, 1/2 so they will be leaving.
January, overall, tends to be somewhat easier to book since many people don't like to take kids out of school for a vacation when they have just returned from the Christmas Break. The exception is the long-weekend of MLK Day.