Not sure if you’ve made this change already but perhaps consider how your son might experience a hyphenated first name as grows into an adult. As suggested above, you may be best to talk to an expert about this as I’m not sure the hyphenated first name will address the issue of perhaps feeling rushed or pressured into finding an alternative to [name_m]Jack[/name_m].
Personally I LOVE the name [name_m]Finn[/name_m], which is a complete, full name, and very much dislike contrived lengthenings. [name_f]Love[/name_f] that you’ve used [name_m]Christian[/name_m] as the middle name as a connection to you - great choice. If you are still really hankering to call your son [name_m]Jack[/name_m], talk to your partner and make clear your feelings. It’s not too late to change, but change it to what you really want rather than adding names or complications that still leave you with an option you’re not really happy with. He should support your preferred change if it’s affecting you greatly.