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Question: How bad is the traffic from the south shore of Massachusetts to Woods Hole on a Saturday morning in June?
Generally speaking, the first few weekends in June, Cape Cod traffic is quite a bit lighter than in July and August. The closer we get to the end of June, the busier it becomes.
The two most likely areas of concern between the South Shore and Woods Hole are: (1) traffic at the bridge coming onto the Cape, and (2) traffic on Rt. 28 to Falmouth/Woods Hole.
This weekend (6/18-19) is predicted to be sunny and warm, so I expect traffic at the bridges will be at least moderate on Saturday morning.
By "moderate" I mean slowed or maybe some stop-and-go just before the bridge, but probably not the 4-5 mile backups that happen on Saturday mornings in July and August.
Also, it's Father's Day weekend, and there's a lot going on all over the Cape - including the popular Arts Alive Festival in Falmouth. So I suspect traffic on Rt. 28 toward Falmouth and Woods Hole will be busy, too.
If I were you, I'd err on the side of caution and get on the road as early as possible. If you're at the bridge by around 8 a.m., you should be okay. Much later than that, you're taking your chances with possible traffic jams approaching the bridges.
I hope this helps. Have a wonderful time on the Cape!!