Yes a WiFi extender would work great, the only kinda bad thing is the extender will have a different set of credentials. This will still let you send files to it, but IMO its kinda a pain. I would highly recommend instead just set up a Mesh Network.
They are super easy to setup and typically come with 3 pucks that you place around the house. I use the Google WiFi and it could not be easier to setup and use. The mobile app tells you if the puck placement is optimal or you can adjust them. The nice thing is that its all one network and the signal is amplified across the house. They also auto update to the latest security updates which is great too.
Typically most of the 3 puck sets cover up to 7,000sqft. But if you find that your garage/shop is still out of range you can add a 4th one close to the common wall to take it over the edge.
If you have a Costco Membership you can get them at great deals.
Just search “Mesh Network” and you’ll have more reading/watching than you can handle. LOL
Here is another good option that is half the price of the Google/Nest ones.