The ticket is linked to a company based on which company the contact is linked to when the ticket was created. Even if the contact is changed to a different company, the same will not reflect on the ticket as the ticket is part of the 'First' company and should not be viewed by the contacts of the 'Second' company.
If you want to change the company the ticket is associated to, then change the requester to a dummy contact and then change it back to the original requester who is associated to the right company.