From what you just described, my recommendation to you as a "novice audiophile" is NOT to run out and buy a different CD player. Of course you may want to get a better CDP at some point, but this will have almost no bearing on your system's low frequency situation.
Listening to a portable through headphones with the bass boost engaged is not the way to judge accurate response, so if that is your subjective preference benchmark, you might be disappointed in this hobby. For better or worse, a speaker like a Thiel is designed to give flat response within its frequency range, so if a recording is bass-shy or if the treble is rolled off, that's the way it's going to sound, and audiophile preamps generally don't have tone controls to compensate. But even if you were to add an equalizer to the system, it could never duplicate the effect of your headphone rig.
As you surmise, the particular Thiel model you own is not designed to go all the way down in the bass (no small speaker does), and also won't exagerate the mid and upper bass instead for added "warmth". It's appropriate for smaller rooms, and listeners willing to live without the bottom 2.5 octaves being reproduced in full, who value its strengths elsewhere. This particular listening style and set of priorities is somewhat peculiar to one branch of the audiophile tree, and also dependent on the music being played. As you probably know, it wouldn't impress most kids who enjoy blasting hip-hop in their cars.
Whether it can give you Ron Carter is the question however. Integrating a sub for satisfactory results can be tricky. The Thiel subs probably would work fantastically with the 1.6, but as you note this seems a ridiculous proposition price-wise. Also, the Thiel subs are designed expressly to extend missing bass frequencies accurately -- not to 'goose' the whole lower end of the spectrum like a bass boost button, or to provide overt "warmth".
[Also, be aware that the word warmth has multiple meanings in audiophilia, not all of them having to do with frequency response per se. Prominent treble is considered to detract from warmth, as is a depressed lower midrange. Deep bass extension, but paradoxically also higher bass harmonic distortion and looser bass driver control, are typically thought of as adding warmth. Another meaning has to do with the harmonic signature imparted by gear in its reproduction of overtones. Anyway, this is why, when you ask about a warmer CDP, you won't necessarily get a solution to bass-shyness.]
What I'm wondering quite honestly is if you bought the wrong speakers. I'd appreciate you describing your room dimensions and listening tastes. Without this info, all you will get is all that you knew to ask for: pet recommendations for CD player shopping, rather than help with your problem.
