What you need is a new Sycamore contact. They allow you to plug boards together with minimal spacing in between. This is especially useful when making add-ons for a basic board which can be replaced and upgraded.
I would advise you to put a via through the edge of the contact pads (if possible) to prevent the connector ripping off the board in high-stress applications. You will be surprised how long they last!
There are several variations for different pin sizes and top/bottom entry:
Here is a video: Sycamore Contact
This will require compatible male pins on your second board. For my application, female pin headers with long male legs will work, but there may come a time when you wish to attach boards without any noticeable protrusions on either board. SMD male pins that I have found so far are not adequate because they leave a gap between boards. Feel free to comment useful part #s below.
I found that Escutcheon pins are a great alternative to pin headers for a protrusion-free connection to the Sycamore contact:
The pins are gold-plated for a longer contact life. Add a thin strip of spongy adhesive between the boards and you're good to go!