Puts waterproof coat on — so Cleopatra survives a little longer? (8)

'puts' is the definition.
The definition and answer can be both to do with contact as well as being verbs in their -s form.
Perhaps you can see a link between them that I don't see?

'waterproof coat on so cleopatra survives a little longer?' is the wordplay.
I cannot really understand how this works, but
'so' could be 'as' (eg 'not as tall as me' means 'not so tall as me') and 'as' is found within the answer.
'a' is found in the answer.
'little' could be 'l' (abbreviation for little) and 'l' is found within the answer.
This explanation may well be incorrect...

Can you help me to learn more?