Should I bother to try a subwoofer?

JL has limited hookup options, especially compared to the Rel's. JL's are great subs, but you really need a certain set up to use them. Also, they do not allow for mulitple input signals as the Rel does. If you have Rel's you can put your LFE signal into one input from your HT processor, and the Speakeron connection from a dedicated 2 channel preamp, and it will work with both signals, the JL will not do that.

Also, the F112 mentioned costs far more than my mid-fi budget allows. I paid 425 + tax for the used REL.

It isn't very likely that your listening position is actually the optimal spot for all sound wavelengths, meaning that you are probably sacrificing sound quality in some frequency range no matter where you're sitting. In many rooms one could find the sweet spot for optimal sound stage and dispersion, but that spot be less than optimal for the longer bass wavelengths. With a sub you have the luxury of obtaining the correct placement for both creation of sound stage and full bass wavelengths. The best of both worlds.

I too am a huge REL fan. Teajay said it correctly in the first post. I bought a Stadium 2 almost 15 years ago and have found it had dramatic improvements in every single pair of speakers I have ever used it with including:
Martin Logan Aerius i's, Audio Physic Tempo 3's,Silverline Sonata 2's, Apogeee Mini Grands, and currently with my Apogee Duetta Signatures.
It is simply the best single component in my system and adds so much improvement I will never listen without its obvious benefits always included. Follow the above advice: crossover pretty low, gain pretty low. I truly can't imagine any speaker not gaining some improvement using a properly integrated sub like a REL, into their room.
I would almost label it "REQUIRED" to get the best out of your system...it's THAT much of an improvement.
Happy Lissn'n

A good subwoofer not only gives you better bass but will increase the size of your soundstage. I could see a major improvement with your speakers and a quality subwoofer. Good luck.