The hunger temptation came first in both narratives, and Matthew says Satan took Jesus up to the Temple, then to a very high mountain. But Luke reverses that order. Or did Luke un-reverse it? Was Matthew the re-arranger because he really likes mountains? Or did Luke arrange things to foreshadow conflict in Jerusalem, at the end? We don't know, but these are all literary questions. Narrative sequence does NOT necessarily imply an historical event sequence.
What about history? Do we even know for sure that the hunger temptation came first? Yes, and not because it opens both narratives. We know this from details in the narratives - especially Matthew's.
"Having fasted..." Matthew says, "he afterwards hungered." But no one needs to read "afterwards" to know hunger follows a fast. What that word does is to clarify that the hunger temptation did not take place during the forty days. That requires some immediacy. In other words, assuming Jesus broke his fast on the 41st day, the hunger temptation would have to occur some time on day 41, before Jesus ate.
We also have Matthew's τότε ('then'). Without going into an in depth adverb study right now, let's just say it is most likely that tote denotes actual event sequence in Mt. 4:5, 10 & 11. The most helpful of these are the last two, where Jesus dismisses the devil (up on the mountaintop) and "then" Satan departs. This comports well enough with Luke's more general "having ended every temptation, the devil departed" (4:13). (This also requires that Matthew's source was aware of the actual sequence, and my own guess is that Jesus was Matthew's direct source.)
So it seems the mountain was last and the bread was indeed first. Still, one question remains. Was the Temple episode "second"?
To assume historicity of the passage is not to assume it has all the details. To assume these three temptations actually happened is not to assume they are all that happened. Maybe Satan was the one who decided on using the the rule of three in his efforts to tempt Jesus, but there could have been more than three - in which case, Jesus or his earliest archivist simply left something out, or Matthew and/or Luke did.
For whatever reason, the Gospel writers only tell us of these three temptations. What they do tell us cannot be all that happened, but it is what we know. Even that much, however, requires some unfolding to get it from its present narrative forms into something more four-dimensional.
Next: Situating Jesus' Temptations