Push-to-open lid for box using a special latch?
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.