Unless you have a standing appointment, you should not need more than five tutorials, and most people find that two can be sufficiently helpful. If you are an experienced and strong writer in your third or fourth year, a single appointment for a complex essay may be all you want. Ask the receptionist when the tutor works and if any of those hours are open. But it may be difficult to make an appointment if you will work only with one tutor. Each tutor is at the Center a limited number of hours, and many tutors have at least half of their hours scheduled with standing appointments.
Most people are just as happy with one tutor as another.
In short, no. We have tried two-hour appointments in the past on an ad hoc basis, but a couple of people did not show up for their long appointments, and many people wanted these long appointments during peak times, such as the week before final exams. Overall, the two-hour appointments inconvenienced more students than they helped. Complete your draft. Outline what you have already written to see if the structure looks logical to you.