Can Another Mortgage Company be Used After the Completed Appraisal?

You can certainly use another mortgage company, however, in most cases you will not be able to use the appraisal you already have.  Each lender selects their own appraisal management company that they have worked with and trust, and are unwilling to use the appraisal of someone else.  

Prior to May, 2009 this was not the case.  So long as the appraiser was certified, a lender would accept the appraisal.  But there were many irregularities with a number of appraisals in the mid-2000's that played a role in the meltdown of the housing market in 2007.  

As a result, the Frank-Dodd Act was passed and it called for appraisals to be done independently of the lender - in fact lenders may not even talk to the appraiser any more.  Since then,  Appraisal Management Companies have been established and lenders have set up alliances with one or more of these companies.  For the most part, they insist on using these companies exclusively.