The next day I woke up at 5:30am and after a quick cab ride I was at the Petra visitor centre at 6am ready for opening. A few others on the tour were keen for a morning visit but didn't end up showing so it was just me.
There were perhaps 5 other tourists with me power-walking down to the Treasury right after opening at 6am and we all made it to the Treasury by 6:25am. Although the early morning light was very similar to the afternoon light in the Treasury, having the place to yourself (almost) was incredible. The lack of horses and camels loudy roaming everywhere also helped significantly to the feel of the space. I walked up to the mid-height viewing platform of the Treasury for a few obligatory selfies (not attached).
I then hiked back up to the High Place of Sacrifice which was truly magical as I had the entire area to myself. By 7:30 I was back at the Treasury and it was busier then I had seen in the days prior! As I walked back up the Siq many more huge tour groups were making there way in.
As such, my recommendation would be to get there at 6am (and no later) or in the afternoon, as by the time I got to the visitor centre there were huge queues at security and the ticket counter. Our guide did say this was close to high season however.
A few parting words regarding Petra and the surrounding area. As part of the relocation deal with the local Bedouin people and the Jordanian government in 1985 when they were relocated from the caves of Petra to nearby village, the locals can still make money off the site. As such there are stalls everywhere selling generic Jordan memorabilia and offers for camel and horse rides throughout the site. There are many children who work in Petra selling items and rides as well rather than going to school. It is a difficult situation for all parties involved but we avoided giving anything to the children as to not support the behaviour. This did detract from the overall experience and hopefully someone smarter than me can come up with a solution.
Also be prepared for A LOT of walking and stairs if you want to cover the entire site (unless you want to support the horse and camel riding). As stated by others 2 days minimum to see everything. Petra is absolutely magical and I am so, so glad I have visited.