I've ordered the parts for the TinyLNA board including the PCB. In the meantime, I will begin constructing a Planar Disk Antenna as is suggested on the Reddit rtl-sdr forum. I've acquired my 12 inch pizza pans from the dollar store, but considering my success with the whip antenna, I'm not sure how much better the performance will be. I can only build it and find out.
In the photo above, there's a run of cable across the bottom pizza pan to a connector. My plan is to solder a female SMA connector between the two pizza pans and then I will have the option of connecting the TinyLNA to it before the cable run back to the RTL-SDR.
However, it may be far simpler to construct a small ADS-B dipole antenna. That would require just two stiff pieces of wire and the female SMA. It wouldn't be that sturdy, but I may be able to use hot glue or something around the connector to provide some strain relief on the solder joints. The instructions on this forum thread seems like a good guide to follow.