Flight Instructor reported while on a training flight they encountered a smoke cloud during a missed approach and decided to alter their flight path to avoid the smoke cloud resulting in flying towards terrain. ATC provided assistance and the aircraft landed safely at their home airport.

Narrative

I; a CFII; and my student were performing the VOR Runway XX Approach into ZZZ in the evening while completing his Part 61 long cross-country flight. We were returning to the area from the north; where we originally departed ZZZ1. My student and I expected; planned; and briefed the VOR XX multiple times; including the missed approach as well. We were cleared for the approach by ZZZ Approach; and from what I remember; and my student recalling as well; was being told to 'do your best to avoid the TFR.'Before taking the flight; I had briefed myself and my student on all of the most current information on the TFR that we had at the time. While going direct to ZZZZZ; my student in VMC conditions (who was using a view-limiting device) was digressing off our direct course; and I was telling him to correct. Approach verified we were going to ZZZZZ and not direct to ZZZ. I confirmed that we were and informed them it was a training flight and one pilot was under foggles while the other could see outside. From there; my student conducted the approach. We switched to the ZZZ CTAF and made callouts for VFR traffic; alerting them of our intentions to overfly the Runway 500 feet above the traffic pattern altitude for traffic as well as noise abatement for the nearby town.I had known the TFR was north of ZZZ; but I was unsure exactly how many miles north. While going missed; I saw how thick the cloud of smoke was directly in front of us and close it was to ZZZ. If we continued to fly the missed approach; we would end up in the thick cloud of smoke. The first things that popped into my head were two concerns: the first being the cardiovascular and respiratory harm that could happen to myself and my student; and the second being possible engine failure or damage and an engine failure while in IMC conditions. With our safety in mind as well as the TFR; I made the decision; as we were in VMC conditions and it was also VMC conditions south and west of ZZZ; to modify my missed approach in a safer manner.Usually; I would make more position and altitude callouts on the CTAF before switching back to Approach frequency; but I wanted to call and inform them we were on a modified miss out of ZZZ. We were alerted by them of low terrain; were vectored quickly; and told to climb. I thank the controller for being so vigilant and on top of our position. This response from them makes me; as a pilot; trust the controllers even more than I already had.I have reviewed this scenario by myself; with my student; and with my chief flight instructor; and have seen my mistakes. Before starting the flight; I should have chosen a VOR approach for my student that did not put us anywhere near a TFR; this would have solved the problem at hand immediately. Secondly; while briefing the approach and in VMC conditions; I should have taken action to coordinate a modified missed approach with the Approach Controller. Another option I discussed with my student is that; as we were in VMC conditions and I had positive visual contact with ZZZ before we were cleared for the approach; I should have canceled our IFR flight plan and proceeded inbound for ZZZ under VFR. Another area that could have been improved on my end was verifying with the controller the missed approach and what she meant by avoiding the TFR.I see where I went wrong; and I will continue to review this situation. I will take what I have learned from this situation to better myself as a pilot and pass my knowledge and talk about my mistakes to my students and other pilots to prevent them from making the same mistakes that I did. I am open to any refresher courses or safety courses to become a better pilot. I also am going to tour an ATC Tower to see what they see on their end when communicating with pilots such as myself and others while they are under IFR.