You can do that or the way I was taught was tie a piece of string on one of the blades and start it, etc..etc...
Clutch fans are cheap, and if they are marginal will give you a problem.
I would check your antifreeze and make sure it's clean. Dirty, old, rusty fluid won't cool as well as clean fluid will. Antifreeze retains heat by nature, so dirty it's just worse.
Take the radiator cap and have it tested or replaced, as a bad one can cause it to over heat as well.
Take the radiator cap off (when it's cold) and start the engine. Watch the fluid and see if you can see it circulate when the engine gets warm and the thermostat opens up. If you can't see this, then the thermostat is probably bad. You can test it by putting it in a pot of cold water, heat up the water, and when the water get's up to tempurature, the thermostat should fully open.
You can also take a house fan, when the engine is overheating, and place it blowing through the radiator. See if it cools off the engine. If it does, then the clutch is slipping.
