You can update the assigned user for contacts directly the list page of the Contacts or Leads modules. To do this, click on the currently assigned user in the Assigned User column, and select the new user to be assigned. 
To update the assigned user of multiple contacts in bulk:
- Go to the Contacts module from the left sidebar.
- Select the contact for whom you want to update the assigned user.
- Click Edit Assigned User above the list of contacts.
- Select the user that you want to assign. (You can also select "Unassign" to remove the assigned user from the select contact.)
- Click Update.
The assigned user for the selected contacts will be updated.