The construction of the new bridge - which opened in October of 1937 - was already underway when the 1937 flood hit. The flood did wipe out one of the approaches to the bridge, but it was quickly replaced. The wooden bridge itself withstood the flood.
The Falmouth Outlook noted on January 29 that “The approach on the north end of the Butler bridge, about 50 feet long, was washed away. It is not thought that the wooden bridge proper is damage.”
“The old wooden bridge at Butler held to its moorings in fine condition. Water crossed approaches at both ends quickly and went over the flood floor of the bridge, but it was built heavy and strong and withstood the river's attack. Piers of the new concrete and steel bridge in process of construction were completely covered by the water.”
The full Outlook 1937 Flood coverage is here.