Here's a bit of a raw brainstorm on many of the different configuration settings ecobee offers. One of the first things I need to do before I can present recommendations is determine what type of system you have. Unfortunately, figuring that out is more difficult than you would think; there's no one box that says something like "Heat: Gas, Cool: Heat Pump".
For example, I have a two stage heat pump for both heating and cooling. Ecobee says:
hasHeatPump: true
coolStages: 2
heatStages: 1
Apparently heatStages actually means furnaceStages and there is no way to know what actually heats and cools my house. Ecobee knows just enough to call for heating/cooling based on the connected wires.