General aviation instructor reported a near miss with another aircraft during the initial climb; when passed by another turbine aircraft taking off behind the instructor at a non-towered airport.

Narrative

Student and instructor in Aircraft X were conducting pattern laps at JNX. While taking the runway during their second-to-last pattern lap; there was Aircraft Y holding short. Once Aircraft X began its takeoff roll; Aircraft Y made a radio call that they were taking the active runway and departing to the west. During the climb; the instructor told the student there was inbound traffic 3 miles out on the 45 for Runway 21; so the instructor called to extend their departure leg for Runway 21. At that point; Aircraft Y made a radio call that they would be passing below Aircraft X. The instructor noted Aircraft Y and Aircraft X were 300 feet apart vertically. The instructor then gained visual on the aircraft once it passed in front. Aircraft X was required to continue the climb on the departure leg due to inbound 45 traffic and could not turn west because that was the direction of flight for Aircraft Y. This could all have been prevented by Aircraft Y holding short of Runway 21 until Aircraft X completed its left crosswind turn; or until it had adequate horizontal distance from Aircraft X.