Single click almost always indicates a failed stater, regardless of starter age. The sound is the spring loaded plunger inside the solenoid moving. The issue is probably that the copper contacts inside the solenoid, that the plunger is supposed to contact, are worn out so no electrical connection is made. If you purchased a rebuilt starter, then this is even more likely to be your issue. I would recommend having a new plunger and contacts installed in the starter, which should be half the price of having a new starter installed.
Best of luck.
Best of luck.