It's likely that Hittite had an /o/ (and represented it in cuneiform!), but most transcriptions still don't reflect this.
In Hittite cuneiform, the signs U and Ú were both used frequently for phonetic purposes. This isn't too weird—Akkadian did this too—and for a long time it was assumed their use was fairly arbitrary, like the distinction between TA and DA in Hittite. They both represent the same sound, and the choice to use one versus the other comes down to the individual scribe and certain conventions.
But more recently, corpus analysis suggests that U and Ú were much less interchangeable than TA and DA. Most words are almost exclusively spelled with one or the other. For example, lūli- "pond" is always spelled lu-ú-li-, and lūri- "disgrace" is almost always spelled lu-u-ri-.
Similarly, while the morpheme u- "hither" is spelled both ways, within a particular verb it's entirely consistent which one is used. The verb ūnna- "drive hither" is always spelled ú-un-na-, while ūššiya- "open (curtains)" is u-uš-ši-ya-.
In that case, which is which? Well, if we compare the paradigm of the ablauting verb au/u- "see" against pai/pi- "give", we see, for example, first singular u-uḫ-ḫi (ūḫḫi with U) and pé-e-eḫ-ḫi (pēḫḫi with E), but first plural ú-me-e-ni (ūmeni with Ú) and pí-i-ú-e-ni (pīweni with I). Kloekhorst considers this conclusive proof that the u~ú alternation happens in the same environments as e~i alternation.
(For some reason, in his argument Kloekhorst uses the spelling pí-ú-e-ni, but that could just as well be read pé-ú-e-ni. I don't know why he doesn't cite the clearer form pí-i-ú-e-ni, which is also attested and is unambiguous.)
Based on this, it seems the U sign was used for /o/, and the Ú sign for /u/. This theory has been gradually catching on, but transcriptions are slow to change, and still write lūri- instead of lōri-. It didn't get its own set of syllabograms—that is, there was no DO sign separate from DU, and so on—but plene spellings could be used to clarify it, just like with the ambiguity between /e/ and /i/. After all, there was no DE sign separate from DI either! (Side note: both ŠU and ŠÚ were commonly used in Hittite. I wonder if they were ever repurposed to write this distinction? I'm guessing not, but that might be an interesting study.)
For more details, see Kloekhorst's Etymological Dictionary of the Hittite Inherited Lexicon; a lot of his phonetic conclusions in that work are controversial to say the least, but he provides a lot of data on the U vs Ú distinction and that part at least seems well-accepted.
