Is it obligatory to continue supplicating for something, despite having previously performed the Istikhara prayer and the matter not being completed?

There is no harm in continuing with the aforementioned supplication, as it is a supplication of istikhara (seeking guidance) that does not conflict with the previous istikhara prayer. One must be content with what Allah has decreed and remember His saying, the Almighty: "And perhaps you dislike a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah Knows, while you know not."