Single-engine piston instructor reported descending towards terrain at night. Flight climbed and continued to destination with no further incident.

Narrative

A student and I filed an IFR flight plan from ZZZ - ZZZ2. We had filed ZZZ COS ZZZZZ ZZZ1 ZZZ2. When we called ZZZ ground to pick up our clearance the controller asked us if we wanted to land in COS and then continue on our IFR flight plan to ZZZ2 (stop and go). We confirmed; yes; and he said he had to modify our flight plan and would give us our clearance shortly. This flight was to complete a lesson in our training which required a long cross-country flight with landings at least three locations which three different types of instrument approaches (hence why we were asking for a landing at COS). After finishing the runup we contacted ground again and he issued our clearance which was the ZZZZZ1 departure and then as filed.We took off with no issue; and was directed over to COS Approach. Upon contacting them they asked if we wanted to fly (straight) back to ZZZ2 and we clarified saying we wanted an approach and landing at COS and then back to ZZZ2. The controller clarified and confirmed what we wanted and then asked for what approach we wanted into the Springs. We asked for the approach and they started vectoring us for that approach. We were established on the approach and they directed us to contact COS Tower so we did. They immediately cleared us for the option. The rest of the approach was uneventful and we came to a full stop on the runway. We stopped for a second or two and then added full power and took off. Once on the upwind we checked back in with COS tower and told them we'd like to continue our flight plan back to ZZZ2. Tower directed us to climb 9000 and left turn to a north heading.Then they asked if we wanted to fly VFR back to ZZZ2. That's when they told us our IFR flight plan had been closed and we were no longer in the system. There was confusion over why or how it got closed. It sounded like the voice changed between who cleared us to land and who we were talking to once we were in the air after the stop and go; and the controlled apologized and said something to the effect of he just got on. He said if we stay on the ground for too long the flight plan automatically closes. Reviewing our radar track; from the moment our wheels touched the ground to the moment we were airborne was less than 1 minute (including ground roll to come to a stop and ground roll to reach Vr). I told the controller over the radio we were barely on the ground for any period of time and we never left the runway. He was apologetic but couldn't help us with an IFR clearance or flight plan anymore since we were no longer in the system.At that point we had climbed to about 9;000 and were about 15 miles north of the COS airport. This is where I made a big mistake. We were now flying VFR and I knew we had the bravo ahead of us; I gave my student headings to fly northwest so we would head towards the designated Denver VFR flyway in case we couldn't get IFR on the way back home; but I also checked and the northbound altitude for the flyway is 7500. I told my student (who was under simulated instrument conditions) to descend and maintain 7500. However; I didn't double check the ground elevation for the area we were in. As we were approaching our altitude my student asked me if that was the altitude we wanted to be at; or if it was too low. I immediately checked the current field elevation which was listed at 7449 and told him to start climbing to 8500. We were at just under 7800 when we reversed the descent and started climbing.While we were descending I was distracted with pulling up the frequency for Flight Service and looking up what equipment codes our plane was equipped with. I have not flown in this specific area extensively and falsely assumed the ground elevation was similar to Denver (~5000 MSL). It was also very dark and no visible ground lights in the area to help distinguish terrain. I also did not have our MFD in terrain awareness mode which would have helped prevent this situation from happening.After starting the climb I was able to contact flight service and file a new IFR flight plan in the air with their assistance. I vectored my student towards the West side of the Denver bravo as we approached Denver airspace in case we weren't going to be able to open our IFR flight plan in time. We contacted Denver Approach and opened our flight plan at 8;000 ft just southwest of APA airport under the bravo shelf and continued home with no incident.