It looks like the match is essentially done and another match should be starting right away. That means Jamie has to get out of the arena soon so the next set of competitors can get a head start: he could tell as he can see Ritsuka slowly walking away from the ladder and giving him a brief glance over her shoulder.

He can see that their time on the arena is over so he goes towards the ladder he had used to climb up to get there earlier and proceeds to climb down as swiftly as he could. Jamie doesn’t bother to use his flying ability to go downstairs: he just needs some time to process the past life memory and the duel as a whole.