M20J pilot reported receiving an altitude alert from ATC during the departure climb.

Narrative

A terrain warning was triggered about 3nm north of the airport during initial climb out after departing XX at ZZZ in IMC after the pilot delayed initiating the left turn due to a distraction caused by an issue with the iPad EFB. Conditions at the time of departure were 700 ovc; tops reported at 3000; wind 120 at 4. The flight from ZZZ to ZZZ1 was expected to be conducted in VFR conditions based on the forecasted weather. Overnight a thin marine layer moved into the area from the coast to the mountains. Upon arrival at ZZZ two days prior to the event ; I inadvertently left my EFB (iPad mini) at the airport. On the morning of the flight; prior to departing from my hotel for the airport; I was able to obtain METAR and basic weather conditions off my phone but not able to review charts and procedures for ZZZ. After being reunited with my EFB I obtained a Leidos briefing for the route and found the chart data to be out of date so began update of the database. During the update process I called 1800wxbrief and obtained a complete brief for the flight. In addition; I called ZZZ Tower to inquire about procedures for obtaining a departure clearance. Tower informed me to request the clearance with ground prior to taxi. After engine start I received ATIS information; wind 120 at 4; Runway XY in use; then contacted Ground with departure request and told to standby. ZZZ responded saying a departure on XY was not possible this day because of traffic (I suspect this was due to ZZZ2 airspace). I advised Tower I was able to depart XX with the reported winds (120 at 4). Tower advised winds were increasing; currently 120 at 10 and gusting. I advised Tower the current winds (tail) were acceptable. With the wind increasing; I felt pressure to expedite my departure. Ground provide taxi instruction to XX via [Taxiway] 1. Once in the runup area I completed basic runup and preflight checks and received departure clearance over Ground. Cleared to ZZZ [VOR] via the departure procedure; Runway XX departure; climb and maintain 5000; if not VFR at 5000 advise Approach. I reviewed the SID; set ZZZ [VOR] 325 radial in VOR 1 and ZZZ1 [VOR] in VOR 2. During this time I noticed there was a lull in the winds; so I was anxious to depart and advised Tower I was prepared for takeoff. In approximately 15 seconds I received takeoff clearance and began takeoff roll. Takeoff was normal; positive rate of climb; gear/flaps retracted and aircraft trimmed for initial climb Tower handed me off to Approach; I contacted ZZZ [Approach] and was surprised how quickly I entered IMC conditions. Approach confirmed radar contact (they may have requested IDENT). I was on runway heading and looked down at my yoke mounted EFB to verify the departure procedure and realized it had reverted to home screen. I was caught off guard; no chart; no SID and in my haste to depart had not properly briefed the procedure. While I was distracted by the EFB situation the controller advised me I needed to turn. At this point I should have advised to the controller of my situation and requested a vector. Later the controller advised me I had triggered a terrain warning and needed to turn. At this point I requested a heading from the controller and was advised to turn to 260. Seconds later I broke out on top of the overcast. The time span from takeoff to VFR conditions on top was approximately 3 minutes. My failure to properly brief the SID prior to takeoff was the primary contributing factor to this deviation. Next is the distraction caused by the iPad issue and resultant delay in executing the left turn. This includes the reliance on a yoke mounted EFB in the form of an iPad mini as the only EFB on board. To improve cockpit document access I will add a second full size iPad to my flight bag. This addition will provide a backup EFB along with a larger viewing area. Another contributing factor was not having a recent IPC (Instrument Proficiency Check). Also; the charting diagram on the chart is at times confusing and would benefit from more distinct airport and runway graphics.