The drip that came back anyway
You did the job right. Shut off the water, took the handle apart, found the worn rubber washer, and put in a new one from the hardware store bin. For a day or two the kitchen faucet was quiet. Then that soft tick, tick, tick started again at the spout, or maybe it moved to a slow weep around the handle instead. It is a common letdown, and it does not mean you did the repair wrong. It usually means the washer was never the only problem.
Why one part rarely fixes it
A compression faucet, the kind still common in older Republic homes built in the 60s through 80s along streets like Elm and Prairie, has more than a rubber washer working to stop water. There is a valve seat underneath it, a smooth ring of metal the washer presses against. Years of hard water, which is typical for this part of southwest Missouri, roughen that seat with mineral buildup. A brand new washer seated against a pitted or corroded surface will never seal fully, no matter how many times you swap it.
There is also the O-ring on the stem, the packing nut, and sometimes the cartridge itself in newer single-handle faucets. Replacing one piece while the others are worn is like putting a new gasket on an old, warped surface. It buys a little time, not a real fix.
More on this from Reading List Why A New Washer Does Not Stop Every Drip.
What our water and weather add to it
Republic sits on well-fed municipal water with enough mineral content that scale builds up steadily inside fixtures, faster than it would in a region with naturally softer water. That scale does not stay in the pipes. It works into faucet threads, valve seats, and aerators, and it gives new rubber parts a rougher surface to seal against.
Then there is the seasonal swing. Ozark winters bring hard freezes followed by quick thaws, and that temperature cycling stresses old galvanized and even some early copper supply lines feeding these fixtures. A faucet that seems to drip only in January, then eases up by April, is often responding to that freeze-thaw flexing, not just to a worn washer. Summer brings its own wear: irrigation and outside hose use push more mineral-heavy water through outdoor spigots, which accelerates seat corrosion on those fixtures faster than people expect.
What you can check before calling anyone
Turn off the water supply and pull the faucet apart again, the same way you did to install the washer. Look closely at the valve seat, the ring the washer presses into. If it looks pitted, scored, or has a rough white crust instead of a smooth surface, that is your answer. A washer cannot compensate for a damaged seat.
While it is apart, check the O-ring on the stem for cracking or flattening, and look inside the faucet body for chalky buildup. If everything looks smooth and clean and the drip still returns, the problem may be a worn cartridge in a single-handle unit, which is a different repair than a washer job entirely.
If your home still has the original two-handle compression faucets from a build in the 70s or earlier, and you are seeing this same drip pattern in more than one fixture, that is usually a sign of general age and mineral wear across the whole plumbing system, not a coincidence.
Where the job stops being a weekend project
A rough or pitted valve seat can sometimes be resurfaced with a seat-dressing tool, but that only works if the seat is replaceable or reachable, and a lot of older Republic homes have faucets where it is not. At that point the choice is between finding an exact replacement seat, which gets harder the older the faucet is, or replacing the whole fixture.
If you have shut off the water, disassembled the faucet, and inspected the seat and O-ring, and the drip still returns within a week, it is time to call a plumber rather than buy a third bag of washers. This is especially true if the shutoff valve under the sink itself is stiff, slow to close, or leaking around its own stem. Those valves are original to the house in a lot of Republic's older neighborhoods, and forcing one that is seized can turn a ten-dollar faucet repair into a burst pipe under the cabinet.