So, I assume that you'll be adding photos at some point..... so until then, I see it like this. For context, I've successfully upgraded quite a few coins including a handful that "gained" five figures in book value. I've also had a tremendous number of failures and I paid a bunch of grading fees along the way. My experience puts me squarely into the category of small-time wannabe. Others do this day in and day out over decades. If any of those guys have ever seen your coin in its current holder most likely they passed it by.
First, understand, digest, and internalize that at least 3 professional graders have already seen and graded the coin in ideal lighting and in ideal circumstances. While there is a possibility that the coin is, in fact, under-graded, it is far more likely that the coin is graded correctly and you're not noticing the same things the graders did. This is especially true if you are not an experienced grader.
Photos do not reveal much about hairlines or luster, so comparing your coin to photos of other coins can only tell you so much. It's also possible that you're comparing your accurately graded coin to over-graded coins. They're out there, and easier to find than under-graded coins.
Without knowing exactly what your coin is (until presumably you post photos), it's relatively easy to get an upgrade when there isn't much change in value. Turning a 1923 MS65 Peace dollar into an MS65+ Peace dollar doesn't even cover the grading fees. If it's nice, they'll probably do it..... "Sure, yeah, why not? Where we going for lunch today?" Turning a 1934-S MS66 Peace dollar into an MS66+ is a whole 'nuther critter. The graders know it, and the company is potentially on the hook if they make a mistake so coins like that are scrutinized more carefully. "I dunno Bob, do you think it's as nice as I think it is? Let's see what George thinks...... Yeah, now that you mention it, the little patch there under the right wing bugs me a bit too."
Finally, sometimes it's just luck and a numbers game. The best graders are the ones most likely to actually give you a bump up if the coin deserves it. If a grader isn't sure, they're going to take the safe route and leave it alone. You might need to submit it a half-dozen times to get it in front of the people who see it the same way you do.