I have the same issue too. I've tried each of these solutions, and they all work temporarily, but then the problem returns once the app is closed and restarted:
@mikeportanova's solution: share an item with the contact
@ozwoz's solution: do a tap-back reaction to an item in the contact's feed
My own previous solution: Click the button to start a new conversation, and find the person's iMessage contact result. This will start a new conversation thread, but it will have all the old conversation in it, and will have the message input field at the bottom.
I hope this problem can be fixed so these workarounds are not necessary!