Check your owner's manual, but the Elantra most likely is not towable (flat) from the factory.
I looked over on
Remco's website, to see if they make a lube pump or something to allow your Elantra to be flat towed. They don't even list an Elantra for the year 2003. For 2002, they only list a manual tranny version, which they say can be towed. For 2004, they list the Elantra, but do not have a lube pump listed for it.
I believe if you really must bring the Elantra with you, you'll have to do so on a dolly or a flatbed.
The Jeep, being that it has a manual transmission, is much more likely to work with flat towing. Still need to check with Jeep, though, as not all manual tranny vehicles can be flat towed.
My suggestion: Get you a dolly or trailer, and 'tolerate' using that to tow the Elantra for now, since thats the vehicle you want to use. In a few years, when one of your vehicles gets old and/or worn out enough you'd be replacing it anyway, you can then replace it with a vehicle that is flat towable from the factory. Then, you can sell the dolly (or trailer) and get some of the $$ back from it, that you can use to get the new vehicle set up to flat tow.
Will and Cheryl
2021 Newmar Baystar 3014 on F53 (7.3 V8) Chassis ("Brook")
2018 Jeep Wrangler JK ("Wilbur")