How can I download apps that did not do so for my new iPhone?

Assuming you are logged in on your new iPhone with the same Apple ID as those apps were originally downloaded with you can open the App Store app.

Tap your name icon in the upper right corner of the App Store->tap Apps->tap My Apps and all of your previous apps will be displayed for you to download again.

However you could have avoided this by restoring your new iPhone from the backup of your old iPhone and those app would have been downloaded automatically (it usually takes awhile depending how many apps)

Finally, if you are using a different Apple ID on your new iPhone all is lost and you will have to download and pay again for every app again. All purchases (even free ones) are tied to the Apple ID they were purchased with.

And last but not least, as sberman points out, if the app no longer exists in the App Store you cannot download it you new iPhone and are out of luck.

Presumably you have updated from an older iPhone running an older version, likely a much older version of iOS.

Apps that the app developers no longer maintain and are therefore no longer in the App Store cannot be used on your new iPhone.

If a developer removes an app from the App Store, you can continue to use it on your device, but you will no longer be able to download it again. When you set app a new device, it will attempt to download the apps from the App Store that you currently have on your device, and if the app is no longer there, it will not show up on the new device.