Push-to-open lid for box using a special latch? Imagine an open topped rectangle box: it has a bottom and four walls. An interior box fits or nests inside that, and, when seated, is flush with the top.
I want to find a latch/device so that I can push down on the top to disengage it and allow the nested box to rise up and reveal a compartment. Pushing it back down will of course relatch it.
The first response that popped into my alleged mind was a push-to-open latch. The link points to a Home Depot product, but there are many sources. Consider the concept of the lowly retractable ball-point pen. One push extends, the next, retracts.

image courtesy of linked site
The image shows a magnetic attractor to mate with a metal plate on the moving part. If your project has sufficient depth, a single latch could be placed in the center of the inner box or be integrated into the moving portion.
There are other designs of push-to-open latches and some may have a lower profile.