Thermostat fan

We have a gas furnace that was tuned up last fall and a new thermostat installed. We like to have our fan running constantly, which it did all winter. Now that we have the AC on, the fan is not running continuously. I've done my research and know that for the fan to run continously, the fan should be put to the on postion. Here is where my issue lies. With the heat on, the fan runs continously with the fan in the auto position, which is NOT correct. When the ac is on, the fan will NOT run continously in either the on or auto position. I then took the thermostat cover off and noticed that the green wire is not attached at all, hard to see in the picture, but there is a green wire way back in there. One, how was the fan even running with this NOT connected? Two, how do I fix this so the fan will work correctly in the on and auto positions. I took the cover off the furnace and verified that all wires, including the green are connected at the furnace. Is the wiring in the picture connected correctly at the thermostat and do I need to just connect the green wire at the thermostat?