My wife drove her car for, mmm, about 10 years, not knowing the rear defogger grid was never plugged in from the factory.
I'm thinking maybe the same thing for your car.
I need to take the panel off where the antenna connections is and the amp too, so I can see what kind of wire I'm dealing with.
I was thinking of getting anther Accent antenna but want to make sure it would work.
You might have to do some soldering, or you could check at Crutchfield dotcom for an adapter, but a window-wire antenna is conventional technology so you have a lot of aftermarket choices for FM, including any Junkyard pull (eg: Many Chrysler vehicles). You will need to drill a hole in the fender (typically) but they all have decent gaskets that should not cause problems, and even should look OEM to a buyer.
AM is another story; it uses a different antenna technology, commonly internal to the receiver. There are aftermarket AM antennas available, some are passive (no need for a power hookup) and even though most are intended for home use, you might be able to hide it somewhere. As AM technology works, you can even use an inductive type (no physical connection to the receiver) which just has to be placed near the factory unit to work.
There are a couple of gotchas with antennas. For starters, there is a danger of overloading the input circuitry of the receiver which will cause distortion. Powered antennas (contain an "amplifier") are more likely to cause this problem, and generally I avoid them, even in the home. They also tend to be noisy and will amplify noise as well as wanted signal. With passive antennas it rarely is a problem; you might say there is no such thing as "too good" of a passive antenna.
With AM, due to the frequency involved, there is a danger that the antenna itself can interact with vehicle electronics, adding noise, which can cause havoc with sensors, CPUs, and instruments. I would keep any AM solution very close to the deck itself to minimize the chance of that happening, as it's probably been taken into account when designing the vehicle's wiring layout.
The correct length for a half-wave FM antenna is around 57 inches long (or high). You can double (not likely your choice) or evenly divide that length (one-half, one-quarter, one-eigth, and so on) but every time you reduce the length you reduce it's efficiency. Most auto antenna (wire type) are quarter-wave, or about 28 inches in height.
The small "fin" types are multi-interface (phone, radio, navigation) and are almost certainly too integrated into the car's electronics to be swappable.
The common wire antennas used on vehicles are a "ground plane" design; it uses the metal fender as a ground plane which improves reception. It's possibly an issue with the ground plane that makes your back window antenna so ineffective. In that case, the ground plane is the actual surface of the road, and both it's distance from the antenna and it's low position relative to the FM broadcast signal (FM travels in a straight line from there to your reception antenna, so the curvature of the earth is conspiring to place that element below the line-of-sight ideal location.
Similarly, the deviation from vertical of your window-glass antenna (technically a folded dipole type) is an issue ... that type worked reasonably well on 70's~80's era GM vehicles as the front windshield is more vertical in the first place, and was also more vertical in those era vehicles than they would be today.
Modern automotive FM/AM radios use an integrated circuit chip that encompasses the entire radio section, and with aftermarket decks they all use more-or-less the same chip, and often a lower-grade chip than OEM radios (radio quality is not a saleable feature in an aftermarket deck), which tend to be pretty good in vehicles. But one wonders if Hyundai economized there and the radio is just not as well designed or used a lower cost chip than other carmakers use. The point being replacing your OEM radio with an aftermarket deck may not improve anything.
It's probably more than you want or need, but the very best aftermarket FM antenna (in my experience) is the Magnum-Dynalab ST-2, but the cost (about $100) and installation might turn you off. It's not sleek at the base like most car antennas, and it's near 57" high. But since they all more-or-less work the same, If I were you I'd be shopping the junkyard and just pulling something that looks the part. Get all the cable with it (you may need to change the connector, and there is no point in extra length of cable).