As a rule, I make sure the reels are very warm so things are less apt to stick together. Setting them on top of a water heater for a couple hours works well.
Remove the screws from the side cover and very slowly, carefully lift it off the reel. I use alcohol and q-tips dipped in it to wash grease off the gears. If there is a ton of grease, I’ll slip it off with the tip of a blade of my pocket knife or an exacto tool knife, then I wash the gears down. When I am done, I have no visible grease inside the case.
The Hot Sauce is a primo product and is what I use. I put a drop where the handle shaft goes into the housing. I put a drop where the drive shaft runs thru the case towards the rotor and bail. I finish the inside with a couple drops directly on the gears, then re-assemble. Depending on the reel brand, I might put a drop where the rotor and the frame meet [where the rotor turns around the outside of the main body]. I finish by putting a drop at each end of the bail. I give the reel about fifty continuous spins and work the bail a couple times, then I lay the reel back on the water heater on a paper towel or two to catch any oil that may want to crawl out of where it was put.
The Hot Sauce has a hyper low viscosity and creep in and coat hard to get to places without needing to take the whole reel apart. I make-up of the oil makes it adhere to those surfaces without becoming gummy or jelling. Its an excellent cold weather lube as well as an all-season lube.