If you want to increase engagement with your Facebook post and use a unique entry point into Messenger, a Comment-to-Messenger campaign is your best bet.
Setting up a Comment-to-Messenger rule on your post allows you to respond to people commenting on the post with a private message, directly in Messenger.
You can set up the rule so it is triggered by specific keywords in a comment, or by any comment in general.
When people reply to the initial message, you can set up the rule to send out a flow that supports your campaign goals.
We recently worked with an artist who has 27K likes on his Facebook page and managed to collect 1.5K emails, 1.4K phone numbers from one Comment-to-Messenger campaign (which got to 94% reach).
You can read the full Case Study here or read a detailed tutorial on how to set up a Comment-to-Messenger campaign.