Can you use one instance of hand washing, say after leaving a beis hakvaros, to cover you for another, say using the bathroom?
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Shulchan Arukh OC 165 rules one can just wash their hands once when wanting to eat bread after just leaving the bathroom. (He recommends a different practice because of issues related to in what order and when to say the various relevant blessings.)
Seemingly your case is parallel.
There is a concept of אין עושין מצות חבילות חבילות: One may not bundle mitzvot (Pesachim 102). The context of this quote from gemara is with reference to the question of whether one may use the same cup of wine for kiddush and for birkat hamazon. The reasoning being that it seems as if one is just trying to shrug off the mitzvot, instead of caring for them properly. Using this logic, you should do two hand washings.
Further reading: http://www.dafdigest.org/pesachim/Pesachim%20102.pdf