June 1 can be early.
Back in 1990, before the days of online weather, I was supposed to get mostly sunny for the Cascades.
I got a foot of wet snow the first weekend in June and almost froze to death.
Fast forward to June 1, 2022 and another June 1 start from Astoria.
Cold and wet for the first two weeks. Nearly froze on June 14th.
In the Pac NW it's called Juneuary. Course, y'all are from Wisconsin, so ...
https://www.cycleblaze.com/journals/viewpoints/good-lord-its-cold/Your mileage may vary. June 2021 was sizzling in the Pacific Northwest.
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I much prefer a route utilizing the historic Lincoln Highway corridor from Chicago to Pitt.
From Plymouth, IN to Wooster, OH you can take super quiet back roads.
There were many different routings of the original Lincoln Highway.
Plus there's all kinds of history and kitsch along the way.
Indiana route from 2016 - might need a bit of updating
I'm not from the Great Lakes, so I had to hit Indiana Dunes.
Plus there are a good set of paved trails from south of Gary to the Dunes.
In eastern Indiana, the little town of Monroeville has excellent facilities for touring cyclists.
Ohio has multiple back road options - including a 1.5 mile brick stretch of the original Lincoln Highway.
(A bit tough to ride, but a great piece of transportation history.)
https://ridewithgps.com/routes/6839082Pic - Old Trail Road between Pierceton and Columbia City
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East of DC, if you plan a Chesapeake Bay crossing on the weekend -
you can take a passenger only ferry from Point Lookout to Smith Island to Crisfield.
Then wonderful riding on the Eastern Shore.
Another option is to take Amtrak from Union Station in DC down to Fredericksburg
then ride the Northern Neck out to Reedville and a ferry to Tangier Island then to Crisfield.
The Tangier connection is more reliable and runs more frequently.
The Bay Bridge sucks for cyclists.
No bike access, no public transportation, no state shuttle.
So you have to pay $40 per person for a private shuttle.
I have been on their case (Maryland DOT) about this for 20 years.
The Annapolis city bus run to Kent Island was dropped about 10 years ago.