Salam Alaikum,
Please note that Tahajjud Prayer for the last 10 Days of Ramadan will start at 03:00 AM enshallah.
That will allow approximately couple hours after Taraweeh as a rest time, and back to Masjed for Tahajjud (03:00 AM) followed by Fajr Prayer with a small break in between.
Maximize your Duaa and Prayers, may Allah accept them all during those blessed days.