You know, I rolled my eyes a little at first too..but now it has me thinking.
In one of my psychology classes we learned about this thing called the Anchoring effect. People tend to be "anchored" in at the first number they hear, and associate it with more or less, big or little, etc from then on. To explain, some psychologist did an experiment with real estate assessments. Two randomly assigned groups of realators assessed an identical house. One group assessed it as significantly more valuable, like upwards of 50K higher, because the psychologists had used larger numbers in a portfolio they were given. The effect showed up by changing insignificant numbers too..the address, the number of people who lived there, etc...the larger the manipulation the larger the assesments got,the more subtle the manipulation the more subtle the effect...but the effect was there. This is why we have such crazy prices...$3.99, $2,990, etc. We are "anchored" on the "two thousand" and, although we see the nines after it, don't associate it as being equal to three thousand.
Anyway, I'm not trying to support her neurosis or anything, but the fact is that yes, there might just be a real difference between a 3.87 and a 3.9, even though it might not be conscious and calculated. I wouldn't worry about either grade, and I wouldn't suggest summer school or anything to make the difference, but theoretically yes, there is a difference, and I would assume it might be larger than 0.03, but only slightly so.
But OP, if you're seriously asking this question, no, you don't need to take any more classes unless you need them to earn your BS. Your GPA is excellent; if it makes you feel better, you can even report it as a 3.9, because if you round to two significant figures, your GPA rounds up. 
