Time turners are a very simple form of time travel. All you need to do is turn the turner some number of turns, which corresponds to the number of hours back you want to go. For example, Hermione had only to turn it three times, and she and Harry had gone backwards three hours. Is there a way to use a time turner to turn back less than an hour?
It seems unlikely, at least given the current design of Time-Turners.
The Pottermore entry for Time-Turners includes this quote from Prof. Saul Croaker, an Unspeakable who worked on time travel:
We have been able to encase single Hour-Reversal Charms, which are unstable and benefit from containment, in small, enchanted hour-glasses that may be worn around a witch or wizard’s neck and revolved according to the number of hours the user wishes to relive.
Given the name of the charm, it sounds unlikely that it could be used for travelling back for fractions of an hour.
It’s probably possible to create different charms that take you back for different units of time (an hour is somewhat arbitrary), but it doesn’t sound like the current crop of Time-Turners would do that.