Yes, it is possible, but not on this tour.
If you'd like to have the full hike, which is hard and requires quite good physical condition and at least some experience in hiking, then simply book the Teide Peak Full trail Private Hiking tour. Private tour is a bit more expensive for obvious reasons. The private Teide hike also includes Teide permits and transportation and hotel pickup. Book Hiking Teide volcano with permit included Tenerife now just with Booking Deposit on TripsPoint