Alternatives to Italian Tirol/Dolomites

Have you been up the Nordkette in Innsbruck? Nearby Hall looked quaint, but we did not have time to visit, so I can't recommend, but hope to go one day.

Are you certain US citizens (I'm assuming you are one) without residence permits or long-stay visas are permitted to travel to Germany at this time? I have no immediate travel plans and have not done research on this subject, but this German governmental document suggests to me that you will not be admitted. The document includes this statement: "Third-country nationals are not permitted to travel for purposes of tourism." Refer to the section under "What travel restrictions apply to border checks for travellers who are not German citizens?"

I have no idea whether there are similar restrictions in Austria, Switzerland and/or Italy; I just checked for Germany.

Continue from Heiligenblut to Lienz. Consider a night or two in Lienz, a beautiful city that reportedly receives the most sun of any city in Austria. Take the cable car up to Zettersfeld for a good view and relatively easy hiking. Take a look at the Defereggen and Virgen valleys to the northwest of Lienz. It is quite a long drive to Switzerland though. Far to the west in Austria you might consider the area around the Silbertal. I concur you continually check whether you will be able to enter Austria, Germany or Switzerland across boarders or particularly from outside the EU. Though restrictions are loosening, they may go back with increases in COVID19.
https://www.foto-webcam.eu/webcam/lienz/
https://www.schengenvisainfo.com/news/three-things-travellers-to-germany-need-to-know-regarding-coronavirus-entry-restrictions/

To add to acraven's post, the U.S. Embassy links for Austria, Switzerland and Germany, if you haven't already used them (apologies if you have):

https://at.usembassy.gov/health-alert-u-s-embassy-vienna-may-14-2020/
https://ch.usembassy.gov/covid-19-info/
https://de.usembassy.gov/covid-19-information/

More for Austria:
https://www.austria.info/en/service-and-facts/coronavirus-information#quarantined-regions

It would be helpful to hear from posters who are residing in and/or are citizens of the countries above but a quick scan of the embassy sites doesn't look as promising as I'd personally bank on for travel by "third country nationals" by mid June. Even if that is possible, Austria might not, for example, lift the COVID test requirement/14-day quarantine requirement for travelers coming by air from outside of the Schengen. As already suggested, i think you need to keep a close eye on developments over the next week and be ready to accept cancellations if the listed countries are not ready to accept travelers from abroad for purposes of tourism.

Austria Seefeld in Tirol or Germany Mittenwald, about 20km from each

Germany Berchtesgaden

Both areas offer multiple half day hikes for all levels.

@mchpp – thank you for the link to the Lienz webcam! It brought back wonderful memories of when my husband and I hiked there, starting in the 1980s. So gorgeous!

Air travel, except for Austrians repatriating, isn't even happening right now. I am very confused how this OP is even going to get to Europe, much less be admitted.

We spent an amazing 10 days located in the Otz Valley, west of Innsbruck, and loved every minute of it. We used the town of Oetz as a base and spent about 5 days hiking in the Otztal, and the rest of the time exploring other areas in the western Tirols. The best hikes (they were all good) were in the southern part of the valley near the Italian border. Obergurgl and Gurgl had a great hike up to a couple glaciers - not difficult, you take the chairlift up and then hike to the glaciers. The other side of the valley and at the end of the road, we took the chair lift up and then hiked up to a mountain hut called Ramolhaus and then back down to the valley bottom. This area is very interesting. Otzi, the Ice Man - the frozen hunter from 3300 BC era, was found on the Austrian/Italian border here. There were a number of marked in situ sites in the valley bottom where there is evidence of ancient grazing activity and dwellings. There are all kinds of 1/2 day - full day hikes all the way up and down the valley. There are a number of ski resorts here - Solden for one - that make access to the high alpine very easy.

Thanks to all for helpful replies. I was focusing on the EU border openings and didn't account for the fact that the Schengen area will likely continue to limit travel by US citizens for some amount of time. Consequently, we have aborted the 2020 Euro vacation with the hope of completing the itinerary in 2021....