If you have multiple contacts accounts, saving a contact to one that is not your default account is a bit of a pain. Furthermore, there's no easy way to move a contact from one account to another on the iPhone, which is why it's very important to specify your default account properly! You can read more on how to do that here.
For now, let's say that you have two contacts accounts and you want to create a new contact in the non-default account.
In my example, I have two accounts: an Office 365 and a Google Apps account. Google Apps is my default, but I'd like to create a new contact in my Office 365 account.
First, head to the Contacts app and tap "groups" at the top left.
In Groups, uncheck all your various accounts aside from the non-default account you'd like to create a new contact in and click "done" at the top right.
Now, when you create a new contact, it'll go in whatever account you left checked rather than the default contacts account.