Since it's a custom build, maybe orient the jack somewhere where a headphone amp can be plugged in without too much trouble? This allows you to use a regular headphone amp.
I noticed that my description may not be very good. Here's are 2 pictures to show what I mean:
In the first picture, the jack is on the body and the amp doesn't cause much trouble when handling or in the guitar stand or even in put into a case, so the amp can be attached 24x7.
In the second picture, the jack is on the bottom of the body; this location means that the amp must be removed when the guitar is not being worn (otherwise, the amp will get crushed if the guitar is put down/on its stand), and definitely, it won't fit into a guitar case. So, this location can't work 24x7.
Since your uke will be fully custom, if you put the jack on the body like picture #1 (perhaps even in a better location since you have full control), you can use an off the shelf headphone amp without needing to embed that stuff into the uke.