My car feels jerking/hiccup a little bit while shifting to 2nd gear

Hello, I think fuel is your problem as well, but that is just an observation. I am concerned that your fuel quality may be poor, or that there has been water intrusion into the fuel. The first thing to do here is scan for trouble codes. Even if the check engine light is not on, there is still a chance that a code is pending, just not set. At the least, you should have a code that says you have a misfire. (P0300-P0306). If the misfire is isolated to one or two cylinders, you can change the coils between cylinders to see if the code changes to the cylinder you moved it to. Also, Hyundai recommends cleaning the fuel injectors with scheduled maintenance, and this may help if fuel is indeed your issue.