854
But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
Also arguably the ring was having a big effect on him the entire time. The entire journey sam hates Smegal, which he sees as what Frodo is becoming carrying the ring. Smegal was a walking personification of the fear sam had for Frodo. That fear was turned into hate by the ring slowly corrupting Sam's love for his master into a weakness for the ring to leverage. He absolutely was not immune to its power.