Both accounts stay linked and both get scanned; the header only names an account when the latest scan found something for it. That's one of two different situations, so check which one is yours first.
One account has matches, the other doesn't
"Links for @username" appears for whichever linked account had higher-commission matches in the latest scan. One account showing means Soco found matches for it and none for the other right now, not that the other account dropped off. The quiet account is still linked and still scanned; it has nothing to display yet.
The account is new and hasn't been scanned yet
Scans run on a schedule. The "Last updated: X minutes ago" line at the top of Link Finder is the last scan's timestamp, and an account you connected after that time hasn't been in a scan yet, so it can't have results.
Force a fresh scan
Pull down on the Link Finder screen to refresh. The "Last updated" timestamp resets, and any new results appear once the scan completes. If you'd rather wait, the automatic daily scan picks the new account up on its next run.
What the "Link another account" message means
"Not the links you're looking for? Link another account" is the empty state for an account with no matches in the latest scan, not an instruction to re-add one you already connected. If the account is already linked, pull down to refresh; adding it again won't change the results.
Next steps
Where these results come from: