Booth's visit to the Surratt Boarding House after the assassination

[wsanto]

A lot of people mention Booth losing his hat upon leaping to the stage. Is that a fact or a supposition? Is that based on statements of the witnesses? Wouldn't his hat have his name inside? Wouldn't it have been recovered as evidence or as a souvenier if left on stage? Where is it?

Perhaps he picked up his hat or held it in his hand as he leapt to the stage and put it back on his head after he mounted his horse.