The best way is to use WFS service, with TIME parameter defining the relevant time range.
Example is shown here (you need to replace <INSTANCE_ID> with your instance ID in the URL of the request).
Note that you can also use MAXCC parameter (maximum cloud coverage) in this call to filter for cloudless data.
Use FEATURE_OFFSET parameter to control the starting point within the returned features and MAXFEATURES parameter - the maximum number of parameter of features to be returned by a single request.
As a result you will get a list of all available scenes for the chosen location in JSON format (or XML if set so). Some of the dates may be duplicated if there are two scenes available in the area. You should simply ignore these duplications.