This is a really good question, and one which I have been considering throughout the day, as well. The initial center location as determined by NHC after Recon made a number of passes was not that impressive, and remains under moderate shear. Meanwhile, where the convection is strongest - a good distance to its southeast - is also just about ideally situated under favorable winds aloft.

Weak tropical cyclones (and one could arguably have seen TD13 going at least another few hours without NHC having officiated it, as it is just that weak) are much more likely to have center reformations. While the location identified by NHC has appeared to be the most prominent, at the time that NHC made the upgrade I could count no less than four surface swirls in the GOM, plus also one just inland over eastern Mexico.

If the location of TD13 does jump, this will call in to question the viability of all of the initial model runs. Models are only going to be as good as the data which goes in to them, and in the case of THIRTEEN, much greater discernment may need to be used.

As can be seen in the image below, both convection and wind barbs support the notion that THIRTEEN may already be in the process of consolidating well to the southeast. The red "L" is roughly the initial location of THIRTEEN as identified by NHC during the first Advisory. The red "NC?" stands for New Center?

On the other hand, this recent ASCAT pass from 0248Z suggests that the original, NHC-declared center, remains largely intact, but has drifted (or reformed) to the south-southwest:

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With Recon scheduled to make another fly into THIRTEEN later tonight, we will hopefully have a better clue as to what the system is doing come daybreak.