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In Jan., I will be starting a PhD in Applied Mathematics at the Colorado School of Mines. However, I was thinking about pursuing a MS in Financial Math afterwards. Would that be complete overkill or a good idea?

Thank you for your time,

wait until you finish your PhD and then decide. Things might be totally different then.

I agree, besides, a Ph.D is quite long, you can take course and self-learn all you'll learn in MS in the meanwhile.
You'll be doing a Ph.D in applied math , financial mathematics easily falls in that category so I see no need to do the same thing twice.
What is your reasoning? Don't you ever want to go out and earn an income?

I do earn an income. I get paid to attend school. My reasoning is that I would think a financial institution might want to see some sort of Mathematical Finance not just a PhD in Math per se.