Innsbruck around May 10 and want to do day hikes

You can catch local buses up into the surrounding villages. It's a much nicer hike downhill than uphill.
When we are in the Innsbruck area, we are always with a rental car. We just love getting up on the mountainside and driving from village to village.

Yes, it’s possible but will be inconvenient compared to a car.

The train and bus hub out of Innsbruck is top notch. The Innsbruck Pass should give you free local transport as well. The nice thing is if you choose well you can get close to a trailhead and hike down to a spot close to access to public transport so you do not have to loop back to get your vehicle.

Seefeld in Tirol - slightly north-west of Innsbruck - provides 200+ hiking routes:
https://www.seefeld.com/en/walking/hikes

We walked a very short way from the main plaza to the funicular which took us up the mountain. We stopped about half way up and had lots of trails available and wonderful "huts" for lunch. Easy peasy!

If it is not too early in the year you could go up the cliff (train) to Seefeld in Tirol and catch the funicular all the way up for all sorts of trails.

If you look for some time outdoors and great scenery- travel the short train ride to Seefeld.
Seefeld and neighboring villages sit on a high plateau above Innsbruck. You are actually in the mountains.
Hundreds of kilometers of marked hiking pathways. walks with guides including nordic walking available too.
Lifts to take you higher in the mountains if you wish.
Seefeld itself a beautiful town with small lake and famous Church that appears in many tourist photos.
This is about a 30 minute train ride from Innsbruck with frequent connections. By the way, I used innsbruck airport taxis for all my recent transfers and they were fantastic.
Highly recommend.

You can get to the entrance of the Stubaital (Stubai Valley) to the south by bus from the Innsbruck Main Train Station in about 20 minutes. To the end and Stubaier Gletscher @75 minutes. You will have to check the Summer Bus schedule for Regional Bus 590 a/b/N and what lifts are running: https://www.stubai.at/en/
https://www.stubaier-gletscher.com/en/
https://www.stubai.at/fileadmin/userdaten/tvb-stubai/dokumente/pdfs/stubai-panorama-sommer.jpg