Small Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Flight Instructor; Flight Crew Instrument
Flight Crew Last 90 Days 100; Flight Crew Total 6800; Flight Crew Type 2500
Situational Awareness
Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Human Factors; Procedure; Chart Or Publication
Departing JAC; [the] SIC (Second in Command) and I were given the clearance of the Geyser 5 departure procedure: NALSI; MLD; then as filed - J158 ZZZ ZZZZZ - climb and maintain 16;000. SIC and I briefed the departure on the ground in JAC and discussed the nearly 180-degree left turn at NALSI and agreed the aircraft would likely start that turn early. We briefed expediting our climb up to 16;000. We didn't anticipate any issues with that departure as the MSA on the departure procedure read 15;000. The highest terrain on the departure read 13;676. Upon departing from Runway 01 and climbing through 14;800 on autopilot; past GIYER and CFCFD on the 017 radial from JAC outbound to NALSI; the aircraft started a left-hand turn; as anticipated; to intercept the course to MLD. This turn took place approximately 10 miles from NALSI. SIC let Salt Lake Center know we were beginning the turn. The Controller said; Ok; do you have the terrain in sight; my minimum vectoring altitude is 15;800 in that area." He then said 'low altitude alert.' We were passing through 15;000 at that point and were about to enter IMC. We told the Controller we were about to enter IMC and he said expedite climb; which we did. The Controller on the next frequency gave us a phone number to call for a potential pilot deviation."
Captain reported receiving a low altitude alert from ATC during climb-out from JAC airport.