Modify your old 1.1 tool in your LMS
This step is the step that will cause your old 1.1 links to start behaving as 1.3 links.
This step is different between each different LMS.
In place editing:
These LMSs let you change the version of an existing tool meaning you can just edit a tool and change it from 1.1 to 1.3.
These LMSs act according to domain matching meaning once an LTI link is clicked the needed LTI tool is searched for according to the domain.