TB-20 Pilot reported on departure turning the wrong direction on the ODP resulting in flight towards terrain. ATC issued a low altitude alert; turn and climb to get the Pilot back on course.

Narrative

Another instrument-rated pilot and I were aiming to regain proficiency as it had been sometime since either of us had practiced approaches. After he had flown as PIC from ZZZ1 to ZZZ; it was my turn and I planned; briefed; and filed an IFR plan to ZZZ2 from there; expecting MVFR conditions.After pre-flight and before startup; I called clearance delivery for my clearance; which consisted of fly ODP (Obstacle Departure Procedure) then radar vectors to ZZZ2; climb to 5000; and contact ZZZ approach on frequency. At this point; I briefed the ODP; set my heading bug to 150 and ZZZ VOR for the radial. After run-up was complete and I was holding short of RWY XX at ZZZ; I then called clearance delivery again for my release; at which point they advised me the departure frequency had changed to another frequency and I was cleared for departure and void if not off in 5 minutes (departure window: local time XA:37 to XA:42).I reset the standby frequency to ZZZ approach and updated my CRAFT clearance notes to reflect the change. With no other traffic in the pattern; I self-announced and took runway XX and turned left to heading 150 (I should have turned right per the ODP). After climbing thru 1200; I made my last call to the CTAF and hailed ZZZ departure on frequency; when they immediately advised me I had a low altitude alert. After I stated I was performing the ODP; they advised I was performing it incorrectly. At this point; they advised an immediate climb to 6000 ft and an emergency vector to 360. After I was above 3000 ft; I was given a phone number to copy for a possible deviation.Contributing factors: Re-briefing procedure after adjusting plans and updating notes: The combination of a last minute change and short (to a newer pilot) void-if-not-off-by time put me in a 'hurried' state; and I should have asked for more time to get setup and re-brief before departing. Text-based ODP: I believe a visualization of the ODP on documents would have helped prevent my mistake. I looked at my IFR Low chart to visualize what I was going to do; and without the hills to the west on the VFR charts; there was nothing immediately obvious to me that turning to the south would be dangerous. Upon taking off and turning; I also saw the hills; but in a lightly loaded high performance aircraft good climb performance it wasn't obvious that terrain would be a factor (I'd also flown ZZZ RWY XX left crosswind departure when flying VFR previously multiple times).