General aviation pilot reported an unknown loss of ATC communications during cruise flight under an IFR flight plan in mountainous terrain; resulting in an ATC altitude alert on the airway; when communication was re-established. The flight climbed to the cleared altitude and continued the flight.

Narrative

I was operating as pilot in command (PIC) of an IFR flight in Aircraft X from MSO to ZZZ; with three passengers aboard. We departed Missoula shortly after XA:00. After departing Missoula VFR; I received my IFR clearance from Spokane Approach during the climb-out and was assigned 13;000 feet. I climbed to and maintained that altitude. Based on preflight planning; I was aware that a climb to 15;000 feet would eventually be required along my route; but I hoped to delay that climb until instructed in order to minimize passenger discomfort at higher altitudes.Spokane Approach instructed me to contact Salt Lake Center on 127.07. I acknowledged the frequency change and believe I switched over to 127.07; but I am not certain whether I made a check-in call. If I did; it may not have been received. I believe I remained on 127.07 for over an hour without realizing that communication had been lost. According to the archive; Salt Lake Center attempted to reach my aircraft on 132.4 approximately five minutes after the frequency change; suggesting I had already transitioned out of the 127.07 sector. One minute later; Spokane Approach attempted to reach me again on 124.9; requesting that I contact Salt Lake Center on 132.4. I likely did not hear this call; as I had already left that frequency.Between XB:06 and XB:09; I received three cell phone calls from the same unknown number. I ignored the first call; thinking it was a junk call. I answered the second call but could not hear the caller due to a low headset battery that had dropped the Bluetooth connection with my phone. I ignored the third call; realizing that answering again would not help. At XB:12; I received a text message advising that I was out of radio contact and needed to call Salt Lake Center on 132.4. I immediately switched frequencies and reestablished communication. The controller instructed me to climb to 15;000 feet due to terrain and advised that an altitude alert had been triggered. I complied promptly. I explained the situation to the controller and appreciated the opportunity to clarify the circumstances.In hindsight; I believe several factors contributed to this inadvertent loss of communication and deviation below MEA:1. This was my first long cross-country flight in this aircraft; which increased workload.2. We departed later than expected and were flying at night under IFR; had we launched earlier; the flight likely would have been conducted VFR in daylight; with lower workload and fewer procedural demands.3. I was engaged in conversation with my passengers; which created additional distractions.4. My iPad was positioned out of direct view; reducing situational awareness and causing me to miss an ATC message sent via ForeFlight.5. My headset battery ran low during the flight; disabling Bluetooth connectivity and preventing effective use of my phone when ATC attempted to contact me.6. The transition from 127.07 to 132.4 occurred over a relatively short distance; and it appears that no controller attempted to reach me on 127.07 before assuming that I had changed frequency to 132.4.In retrospect; I recognize the need to more deliberately verify two-way communication following frequency changes; especially in unfamiliar airspace. I also plan to make improvements to cockpit organization; ensure backup power and mounting solutions for key devices; and more actively manage cockpit workload during cruise.