A different setup may give you a better sound but that depends on what you get. In a nut shell your pedal is acting as an A/D converter. In other words its taking you analog guitar output and converting it to a digitized wave form. It does say the unit has 96K sampling (although quality will also be had with bit depth) so it does presumably sample at a high rate. If you were looking to step up your game you could go with a 192K converter but I cant say I can hear a huge difference with them.
A DI box will not solve the problem as you still need an interface to digitize the signal. You may see some benefit from using a nice pre-amp with a digital interface like a Universal Audio Solo (or something similar) and a Motu Ultralight or the like.
To answer your question, yes your set up is "sufficient" but there are lots of other options out there to improve it.
Then there is the whole world of putting a mic in front of a guitar amp, but I one can go on for days about that...
A Note on Speakers:
One often overlooked aspect of a small first studio rig is the speakers. Its worth noting that the playback methods are just as important as the recording methods and you may be getting a great tone coming in and never know it if you cant play it back in high quality through a good set of monitors.
In response to your question. Along with amplification your pedal is also Digitizing the signal which is a key operation in the chain. Running a pre amp into the pedal you have wont give you that much. You will gain a lot of quality from a higher end converter and a preamp pair. If you are going to spend the money in one place I would buy a better AD/DA unit to start. There are quite a few on the market and that is worthy of its own question. In short it will give you not only better converters but a few more outs (for monitors and headphones) and maybe even a mic pre to two.