Hey there RichM.,
Thanks for reaching out to the community.
We understand you're trying to change the alarm sound for the wake-up alarm. If you're using the Sleep feature and adjusting that particular alarm, you will only have the sounds provided when you tap Sounds & Haptics. While the resource we're attaching does state it's for an iPod, the information under "Change the next wake up alarm" is still relevant to the iPhone you are using:
Set an alarm on iPhone - Apple Support
If you are trying to set an alarm through the Clock app and are not using the Sleep feature, then you will have other options for alarm tones. These will be found by opening that alarm time and selecting "Sound". You can choose songs, purchased tones, or the tones provided.
This resource will help more with the regular alarms through the Clock app: How to set and change alarms on your iPhone - Apple Support
Take care!