Transmission won`t go into reverse

https://www.autoserviceprofessional.com/article/93110/hyundai-trans-temp
https://static.nhtsa.gov/odi/tsbs/2014/SB-10059669-2273.pdf

https://www.hyundai-forums.com/threads/transmission-fluid-flush-and-or-sensor-replacement.537993/

https://www.hyundai-forums.com/threads/bad-transmission-fluid-temp-sensor.130014/
http://www.carproblemzoo.com/hyundai/sonata/transmission-sensor-problems.php

Check your cable at the trans. And see if it has not loosen up and is not seat right.

For car not changing gears and not shifting out of park mode ,it can be shifter assembly issue, or it can also be transmission issue. Get the shifter assembly inspected with ignition OFF.

For more troubleshooting, I suggest you to get the Following help link checked, its complete step by step procedure to inspect gear shift issues: -----

GEAR SHIFT ISSUES

http://alltypesofpartsreplacing.blogspot.com/2012/11/gear-shift-issues.html

==========

NO REVERSE ON TRANSMISSION
http://alltypesofpartsreplacing.blogspot.com/2012/12/no-reverse-on-transmission.html
-------------

Transmission fluid leak, transmission gears slip,
http://alltypesofpartsreplacing.blogspot.com/2012/12/transmission-fluid-leak-transmission.html
----------

These will help.

Thanks.

maybe theres not enough trans fluid in the trans or the trans is stuffed

make sure that you have the proper amount of transmission fluid. If the tran still has issues then it could just be that the gears on the inside of the transmission need to be rebuilt or replaced.

The battery being disconnected at one point, has nothing to do with
your trans issue. It may just seem like a coincidence, that the trans
just started to act up, once the battery was reconnected.First off, how
many miles does the car have, and have you checked the trans fluid
anytime rescently, to see if it's full?

Did you check the transmission fluid level? If necessary, add fluid. However you probably have a more serious problem. Take the car to a qualified transmission mechanic.

DTC P0734 is "4th gear incorrect ratio" - that's your torque converter 'LOCK UP Clutch" It's worn out (or you're using the wrong Trans Fluid required by Hyundai - SPIII OR Both). The P1624 is simply the Trans Control Unit (computer) communicating to the Engine Control Module (ECM computer) MIL - on request signal to ECM = asking it to turn on "Check Engine" light. MIL = Malfunction Indicator Lamp. When the torque converter 'lock up" clutch slips or doesn't engage, the trans automatically returns to 3rd gear and stays there (it's called 'fail safe' mode) until you cycle the ignition key, then everything resets until the malfunction occurs again.
You'll need to have an AT (Automatic Transaxel) tester hooked up to verify this. Hyundai has a Service Bulletin specifically for "gear incorrect ratio' codes. If when you changed the trans fluid, was it done at a Hyundai Service Center? If not, someone may have installed the wrong fluid.

I don't know if I can help but as an owner of a Hyundai Elantra 2000 GLS with 205,000 miles and the original clutch and transmission I'll try and help!

I always start with the lowest cost solution first and avoid mechanics as much as possible!

Have you checked the trans fluid?
It should be changed every 30,000 miles!
Sometimes a Lucas Trans Fluid will cure - fix slippage!

Have you checked for a clutch adjustmet!
An adjustment can cure - fix slippage!

Are you a clutch rider with your foot always on the clutch?
If so you may have worn out the clutch!

How many miles to you have on your Hyundai Elantra 2000?

i had to put a new trans in my 2000 jimmy about a year ago there is something inside called a "reverse hub" its made from aluminum and not very hard to break..when mine broke i was stuck trying to move in reverse and it heard a pop then a winding down noise and sure enough its the reverse hub to fix it i just had the trans rebuilt