It looks like some of your old site links are there in the database. Thats why it is redirecting to the users to the old site.
My suggestion would be use this
https://interconnectit.com/products/search-and-replace-for-wordpress-databases/
Its a kind of php script. Be sure to keep the database backup before using it. This script will make search and replace the strings from the database. After that just update your permalinks.
Hope this will help you.