Hi there sof513,
It sounds like you're unable to view your Activity awards in the Activity app on your iPhone 6s. We're glad you're here, and we can help.
To start, you mentioned you used a "handoff-like option" to set up the iPhone 6s from your iPhone SE. To confirm, did it look like this process: Use Quick Start to transfer data to a new iPhone, iPad, or iPod touch. If so, did you then restore the rest of your data from an iCloud backup?
In order for Activity and Health related data to transfer to a new iPhone, it would need to be backed up via encryption, and then restored to the new iPhone. iCloud backups are automatically encrypted, but if you backup via a computer, you need to enable encryption for that type of backup: About encrypted backups on your iPhone, iPad, or iPod touch
When transferring from one iPhone to another, and pairing your Apple Watch with a different iPhone, you should follow the steps in this article: How to pair your Apple Watch with a new iPhone. This will ensure your Apple Watch is paired and data is properly transferred between the devices. Step 2 in that article, talks about making sure your Health and Activity settings are backed up prior to transferring to a new device.
Let us know if that helps.
Cheers.