That one click noise is a classic sign of power not reaching the starter. It can come from corroded battery cable connections, or a bad starter relay, or a bad starter solenoid. Corroded battery cable connections often look just fine to the eye; so you cannot tell by looking at them. This is why they have to be properly cleaned. The best way to deal with that is to buy a tapered reamer type battery cable service tool, disconnect and clean both battery cables and both battery posts until there is a shiny ring around the surface. Then install the cable clamps on the battery and tighten them until they cannot be moved by hand. It is also a good idea to remove and clean the metal surface of the other ends of both cables, and securely tighten them.
There is also an inexpensive part on your car called a starter relay. If this part fails, it can cause intermittent starting such as you experienced. Posted below is a link to a photo of this relay; it just plugs in to the underhood or under dash fuse block.
There is a cylindrical part called a starter solenoid which mounts on top of the starter. This part is often not replaced on rebuilt starters; but it fails more often than the starter does. If the starter you bought did not come with a new solenoid, and your starting problems continue; I would definitely buy and install a new starter solenoid.
Many cheap rebuilt starters turn out to be defective. Thay are typically only repaired in the least expensive way; which leaves other parts that should have been replaced vulnerable to failure. After becoming thoroughly disgusted with the unreliablity of cheap rebuilt starters, I now will not buy a rebuilt starter from anywhere but a NAPA parts store. And even there, I'll make sure it is from their premium line of parts.