Hi,
something basic that I have done in the past is have the sketch output the contents of the SD Card in the set up function. As a serial connection resets an UNO anyway, it will run setup and output the logged data automatically everytime it is connected to the serial monitor. It only does this for new connections, so you can leave the sketch running and logging data after the initial connection.
This is how I transfer logged data from my RC cars to the PC, by running the UNO in the car, then after a session I connect a USB cable start the serial monitor and upload the data.
Very simple and does what you want.
Duane B