Communication Breakdown
ATC Issue All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Issued New Clearance
Procedure
Departing HTO. The clearance given was HTO HVN CMH V23 SAX Direct TEB. There is a temporary Tower located at HTO during the summer. It is confusing who actually controls the airspace in the area. Is it NY or the temporary Tower. When departing Runway 10; we were cleared for takeoff without further instructions. We climbed on runway heading to 2000' FT. The HTO Tower switched us to NY on 125.97 and we checked in with NY. After checking in with NY; the Controller ask if we were in the turn to HTO to which I replied we were runway heading. NY then asked if we were cleared direct to HTO. I conferred with the pilot flying who had picked up the clearance and he said that the clearance was to HTO as our 1st way point but that we did not receive instructions to turn direct to HTO after departure. When the Tower Controller cleared  us for takeoff; that was all he said; cleared for takeoff Runway 10. There was no altitude deviation or anything inappropriate; just misleading instructions. The NY Controller queried us about the phraseology used and we told him our instructions given by the Tower Controller. Since the Tower Controller just cleared us for takeoff without further instructions for a turn or OD procedure; we flew the runway heading. That was what we felt the proper procedure was per the takeoff instructions. The flight continued to TEB without further incident. Suggestions; the airspace and the controlling agency of the airspace in and around HTO has to be clarified better. I believe this is a private Tower set up for the summer months only. Proper phraseology needs to be given along with the proper instructions. Neither one of us felt that what we performed during the takeoff was incorrect and given the takeoff instructions; we performed the departure the way we understood it to be. If we were to turn direct to the HTO VOR; the Tower Controller should have phrased it that way when clearing us for takeoff. In reviewing the clearance with the flying pilot during and after the event; given the takeoff phraseology; we would probably perform the departure the same way again.
IFR departure from HTO reported confusion regarding the differing instructions issued by an apparent 'temporary' Tower and N90. The reporter indicated increased clarity is necessary regarding ATC service in the area.