More likely, discontented Democrats will switch to the Republicans, which will change both parties for the worse. A lot will depend on who the Democrats nominate in 1940. If someone like Garner or Byrnes gets the nomination, progressive Democrats will leave for the Republicans and take them over. If someone like Bankhead or Barkley gets the nomination, conservative Democrats will leave for the Republicans.
I actually think the most likely democratic nominee in 1940 will be Cordell Hull as he can unite the party better than any other nominee and he will probably easily defeat whoever the GOP have left which probably will be someone like Charles McNary .
I think though it’s more likely that the pro new deal wing wins out by far given how popular FDR is and he will basically be the kingmaker at the 1940 DNC . So yah probably the anti new deal faction will leave the party at some point before 1942 imo but the question is would they join the GOP as it is or would they merge with them forming a new party given how unpopular the GOP brand is especially in the places they are from(the south ).