You can use muriatic acid as opposed to a coat hanger. Turn off the water supply to the tank and flush it. Plunge/sponge out the remaining bowl water. Then lift or remove the flapper in the tank, pour about two cups of the acid into the hole in the center bottom of the toilet tank and about a quart into the toilet bowl. Wear rubber gloves and be very careful. Turn on the exhaust fan, open the window and leave the premises for at least a half hour. Turn the water back on and it should flush fine if obstructed rim holes and syphon jet are the problem.
In-tank cleaners are nothing but trouble. Besides clogging rim holes, they cause premature deterioration of the tank components. If old-fashioned toilet scrubbing is not your thing, there are several newer products that hang on the side of the bowl that do a decent job. Fluidmaster now makes an in-tank cleaning system that injects the chemicals directly into the bowl so that they don't come into contact with the tank water.