Put on the wooden overcoat
The 'wooden overcoat' is, of course, the coffin. There were variants on this theme: 'the concrete overcoat', 'the Chicago overcoat'. These latter two presumably coined in reference to the murderous methods of the Mafia.
'Wooden overcoat' isn't recorded in many reference works and the origin is uncertain. It appears to be American in origin and most of the early printed citations come from the USA. The earliest that I can find is this piece from the Ohio newspaper The Coshocton Democrat, May 1870:
"If you want to get blowed up - if your life is insured in your wife's name and you want her to have the money immediately - buy some new-fangled burning fluid that is remarkable for its cheapness. There is no surer way of getting yourself into a wooden overcoat."
The 'new-fangled burning fluid referred to was petrol. No longer quite as cheap as it was but still just as flammable.