The south, pretty much the south half of continental USA would be a good place to avoid from Jun through September.
I say that with a very broad brush, as the eastern half of USA in the south will be much more humid than the west.
Are you familiar with dew point temperature and how that relates to how oppressive heat is?
https://www.weather.gov/arx/why_dewpoint_vs_humidityhttps://en.wikipedia.org/wiki/Dew_pointOn this forum I usually give SI units since most people here use those instead of the units common in USA. But if you are in USA,you will have to become used to temperatures in Fahrenheit, so I am using that in this post.
I avoid strong exertion when dewpoints are in the 70s or higher. Dewpoint in the 60s, I sweat a lot on a bike, require more water, but I can perform well. Dewpoints in the 50s or lower are quite comfortable for me, require less water.
And of course, the raw temperature comes into play, even in dry conditions (dewpoints in the 50s or lower) if the temperature is in the 90s or more, that will impair my performance on a bike. Ideal is temp in the 70s and dewpoint in the 50s.
I assume you are familiar with this site, but I mention in case you are not. This is my go to site for planning a trip and the likely weather I will encounter.
https://weatherspark.com/You can pick a location, ideal is one with an airport for past weather data, and enter that location into the website. I live in Madison Wisc, so I will use that:
https://weatherspark.com/y/12796/Average-Weather-in-Madison-Wisconsin-United-States-Year-RoundIt is possible that when you are in Europe it lists the temperature in SI units (C) simply because it assumes you use those units, but the Madison site that I posted above lists the temperatures for me in F.
If you scroll down to the chart titled Humidity Comfort Levels in Madison, the color coding is based on dewpoint.
That website calls precipitation in a day of less than 1mm as not having had any precipitation, so the chart with probability of precipitation on any given day will slightly undercount actual. That chart basically says that the months of July and first half of August have a change of being Oppressive or worse over 10 percent of the time. That is not too bad, you can stop for a day if you feel like it. So, I think bike touring in my area during any time in summer would not be bad.
I assume you already have a good feel for raw temperatures where you like to bike and where you feel it is too hot, so I am not discussing the second chart that shows average temperatures. That of course is important, I just choose not to put much emphasis on it here since I suspect you already are well versed in that.
This past April, I did a tour from northern Mississippi to western Tennesee. Mid-point is roughly Tupelo Mississippi.
https://weatherspark.com/y/146447/Average-Weather-at-Tupelo-Regional-Airport-Mississippi-United-States-Year-RoundIn Tupelo in April, average high temperatures are in the 70s, average low is in the 50s to 60s, and the humidity chart says it will be mostly dry but occasionally comfortable to humid.
But looking at those temperature and humidity charts, July and Aug would be hot and also often oppressive humidity. In other words, avoid that area during that time of the year.
I can't draw a line on a map and say when to go where, you will have to do that, but this website gives you a tool you can use. That site also gives you good info for length of daylight hours, average wind direction and average wind speed. I did that April trip with a friend. In planning he was most insistent that we ride from northeast to southwest, I insisted on southwest to northeast. If you look at the wind direction chart in April, 60 percent of the time we could expect a south or west wind, so I won the argument.
My two examples above (Madison and Tupelo) were in the eastern part of the country that is more humid in summer. If you look at Denver Colorado:
https://weatherspark.com/y/145689/Average-Weather-at-Denver-International-Airport-Colorado-United-States-Year-RoundIt is quite hot in summer in July, but the humidity is low enough that you should be able to ride without problem.
I probably emphasized dew point more than most people. The reason that I focus so heavily on that is that dew point tells you approximately how cool an object will be if you wrapped a damp cloth around it. It could be cooled down to roughly that temperature. (Technically wet bulb temp and dewpoint are close, but not perfectly the same.) Your skin is normally close to 70 degrees. So, when you sweat, if the dewpoint is well below 70 degrees, the evaporative cooling from your sweat will cool you down very well. But if the dew point is 70 degrees, your sweat has to be very efficient to cool your skin down to 70 degrees. If the dew point is 80 degrees, sweating would not be able to cool your skin lower than 80 degrees, which is much hotter than normal for your skin temperature.
One rule of thumb might be to tour in areas where ACA is touring during that time of year. They will pick the best times to tour in various locations.
https://www.adventurecycling.org/guided-tours/On that list of tours, they listed Natchez Trace in March, April, and October. My example with Tupelo location in this e-mail was for my ride this past April on Natchez Trace. (I did not do it with ACA, but we saw them when we were there.) So, ACA would not try to tour there in June through August, but May and September may be borderline months were you could do it but it would be hotter than ideal.
I spent more time writing this than I planned. I hope it helps.
ACA also publishes bike maps for recommended routes and often those maps are for locations other than where they operate tours. Some people find them to be very useful, I have not found them to be that great, but I will agree that their routes are very good for biking. Maps are not cheap.
Would you plan to buy a local (USA) sim card? I do not know if your European phone would have the right frequencies in USA. Perhaps others would know that better than me. I use Puretalk for my phone. They have a plan that is $20 a month (plus taxes on top of that) for both talk and a few gb of data, you can cancel at end of trip. That company uses the ATT network. My phone is 4g and works well, you do not need a 5g phone ... yet.
https://www.puretalk.com/cell-plansA side note on USA and prices. In Europe, when you are quoted a price on something, that price includes VAT tax. In USA, there often is a local or state sales tax that is not included when you are quoted a price. So, each time you buy something, expect a small surprise charge when you go to pay for it. In my community the sales tax is 5.5 percent but it often is higher elsewhere. Some things are generally not taxed, such as most foods in grocery stores, but each local will have different categories of what is taxed or not.
Some western states in USA have had forest fire problems in mid summer in past few years. I won't elaborate more but just comment that it is something to be aware of.
