"rein" is a colloquial short form for "hinein" or "herein". The full verb here is "hineinlegen" (or colloquially: "reinlegen"), meaning "to put into". The adverbs "ganz unten" specify that you put the scarf and gloves at the very bottom of the recipient.
Note, that when you use "hereinlegen" with a person (jemanden hereinlegen), it means "to play a trick on somebody". I am not quite sure where this metaphoric meaning comes from, but it is quite common.