It's really going to depend on the queries that you want to optimize. I think the best help might be to point you to a book: https://www.amazon.com/Foundations-Multidimensional-Structur...
An RTree is the first structure that comes to mind, but the way you describe the problem, it sounds like the intervals never overlap, so I have my doubts. Sounds like you might be looking to optimize the query "what is the first interval of at least N days?" Maybe look at priority trees. They're good at queries that are bounded in one dimension and half-open in the other.