upgrading to iOS 10 on a 3rd generation iPad

You can't, it only supports up to iOS 9.3.5 (9.3.6 if it's cellular).

If the app's developer supports you might be able to download an older compatible version via the Purchased tab in the iPad's App Store app. Is the app already in the account's purchase history, and if it is what happens when you try that ?

What app are you trying to get for your iPad 3 on iOS 9.3.5?

If this is an app you never downloaded previously, there maybe one more way to attempt to get an older, compatible version, but NOT all apps in the iOS App Store have older version to still download .

The decision to leave older versions of apps in the iOS App Store servers is entirely up to the developers of the apps.

Thank you for the helpful but you didn't answer my query?

What app/s are you trying to get?

Is this an app you have had previously?

You cannot use the iOS App Store Purchase history method to get a compatible version of an app if you never had the app to begin with.

Do you own a real computer?

Are you interested in using my other method using an older version of iTunes on a computer?