Yes, the same way that lieutenant commanders can wear two different insignia, if TOS season 1 is taken literally. Either way it is open to be retconned: Either Tormolen is a full lieutenant and "really" wears a full stripe, or the insignia represents a different rank, as it was apparently intended to do, and Tormolen's dialogue is wrong. I prefer the latter, as it fits with the insignia system and TNG better. YMMV.
But I'm not talking about individual preference, or about what you personally imagine to be the case. I'm talking about what there's objective evidence for. The whole point is that the show never clearly establishes such a rank, which is why it's left to individual opinion, and why there was no fan consensus that such a rank existed before TNG made it explicit. If you have to rely on personal belief or extrapolation to arrive at your conclusion, that merely proves my point that there's a gap in the canon itself.
But it seems to me that if they had intended a lt. j.g. rank to be in use in Starfleet, then we would've seen it used repeatedly, not just one ambiguous time. After all, there were dozens of lieutenants seen over the course of the show -- I estimate more than 50 named lieutenants throughout TOS, more than 60 if you count TAS, and that's not even counting unnamed extras. And as a rule, there are fewer officers holding each successively higher rank, so logically there should be more j.g. lieutenants than full lieutenants. But instead there's exactly one apparent instance out of 50 or 60, which is statistically indistinguishable from zero. Which, to me, makes it very unlikely that the producers intended the rank to exist. Even if someone involved in costuming on "The Naked Time" thought it existed, that idea evidently didn't survive the process of trial and error that shaped the conventions of Starfleet as the series went along.