Can't get wrong sender name from appearing -- only when sending from iphone
Maybe I'm missing something or misunderstand the question, but it seems to me the reason for this has nothing to do with your iPhone. Instead, it's most likely related to where the email address is recorded in the contacts on your laptop and on devices belonging to other people.

I'll try and clarify my answer with an example scenario:


Let's say an email address is used by Fred Flintsone. You and others add Fred Flintstone to your various Contact lists on your various devices and add the email address to that contact.
Fred Flintstone retires and then Pebbles Flintstone takes over and assumes the same email address. She uses an iPhone and adds the email account to her settings and uses her name. So now the email address, on her iPhone, appears with her name.
However, Pebbles also uses the laptop that Fred used previously. In this laptop the email address is still listed under Fred's details in the contacts. So, if an email comes into the inbox on this laptop, it appears as though it's come from Fred Flintstone because that's who the email is associated with.
Likewise, if anyone else still has that email address associated with Fred Flintstone in their contact lists, the same thing will happen.


Assuming that your scenario is something like the above, then the answer is to ensure that the email address is associated only with you, not with your former assistant and not with both you and the former assistant on their contact lists.

NOTE: This scenario does not apply in all scenarios (e.g. with a MS Exchange account, etc) but does apply in most situations.