The date a Tweet is posted is displayed in the “created_at” field, as explained here: https://dev.twitter.com/docs/platform-objects/tweets
If you want to get Tweets between two dates which still exist in our search API range (6-9 days into the past) then you can use a query parameter which uses the since: and until: operators. For example, Tweets containing “cats” between August 5 and August 10: “since:2013-08-05 until:2013-08-10 cats”
Beyond that, you’ll have to find a timeline of Tweets you want to look through and perform a manual search for ones which occur between your desired dates, again using the “created_at” attribute.
https://dev.twitter.com/docs/api/1.1/get/search/tweets