The Search column in the Advanced Link configuration table informs whether the Advanced Link is searchable via JQL with or tags.
To activate search for the Advanced Link:
Go to Apps → STAGIL Assets → Advanced Links.
Find the Advanced Links → 3-dot Actions menu → Enable search.
Open the Actions menu again → Reindex.
To disable search for the Advanced Link:
Go to Apps → STAGIL Assets → Advanced Links.
Find the Advanced Links → 3-dot Actions menu → Disable search.