I want to track my phone without using find my phone

There are third party remote device tracking programs in the App Store. I have no experience with them as I don’t see the point when find my iphone is baked into iOS and my iCloud account, but they are there if you simply search the store for them.

I assume you have to leave them running all the time with background permissions to be any use, so they may be power hogs relative to find my iphone, which is highly optimized to keep power use to a minimum. Most seem to be run as services, so you will have to pay to use them as there has to be some form of server that the tracking app sends its data to so you can access that location information.

Note though, that if you are not using find my iphone, then your device is also not protected by activation lock either, as the two are tied together.

You can't.

...but there is no other option for tracking your phone?

You are welcome.