You want autosomal DNA because that is where you get ethnicity estimates.
If, as ColeValleyGirl points out, this ancestor is your father's mother's mother's mother, then his mtDNA (along with any of his siblings') will be the same. Your mtDNA will be your mother's, so not helpful. If that is the case and the mtDNA is one that is from the ethnic group you are looking for, and not in the ethnic group you mostly are, you have your evidence. But mtDNA (and Y-DNA) can be tricky. If your GG grandmother is, say, 90% the group you think, she still might have mtDNA that doesn't match it.
Instead of taking more tests, use the data you have and upload it to other places with good ancestral calculators. The best one is Gedmatch.com. It has multiple calculators you can use, each with many variations. You can also upload to FTDNA for a small fee (they change which 23andme tests they take and also the fees on a regular basis).
If your GG grandmother was truly 100% from the other group (and we'll say her group and yours are very different so there isn't overlap...so not Italian vs Russian but European vs African, or something), on average a GG grandchild would inherit 6.25% of her DNA. This is not that small but the problem is it's just an average. The range of possible DNA can go down to zero. Also, you might have inherited a lot of her DNA but not any of the markers that the companies use to show ethnicity.
Then there is the issue that a lot of people said to be of a certain race/ethnicity in the past actually turn out to be admixed. In other words, even if the family story is true, she may not be 100% whatever. She could be 50% or 25% from intermarriage of her parents and/or grandparents. Or the admixture could be from a while back and not even considered by the society at the time.
Some articles to check out: