If you are looking to add an existing Client Contact and associate them with a new or different Client, you must first change the Client Contact's existing company.
Contacts > Find your Contact's Name > Click it to access the record
On the General tab of the Client Contact record you can change where the contact works using the Client field.
Use the lookup field to change to the proper Client/Company name:
If the Client does not exist, simply click on the +New Client at the bottom, highlighted in blue.
Click on Save and Close at the top of the page.
The Client Contact will now be associated to the Client.