It's difficult to handle that much road traffic in congested areas, for sure. The game isn't perfectly designed to model road transport: you can only have one vehicle per tile, etc.
I'd suggest making your main intersections more robust in order to handle the traffic volume. Using under/over passes, expressways, elevated roads, etc.
Alternate wayfinding is an interesting idea, though I suspect that in most cases the next destination in the route is still through the same traffic jam.