Well worth a check to see if any family members ever applied for Poor Relief.
Where the detailed reports of the Poor Relief Inspector survive they can often give a mini life history, including detailed info on the place of birth, as well as the names of parents, even sometimes names of the families of children and occupations, all of course to establish the extent to which famly members could/should to be able to support the applicant.
These records survive, mostly, for Glasgow. I believe that there are surviving records for Paisley, but I have no experience of the content. Greenock's records, I think, were destroyed by enemy action during WWII.
And there are surviving records in many other places in Scotland.
For someone from Ireland, these records can often be the only place where the exact origin, down to the level of the townland, can be found.
There was no shame involved in applying for Poor Relief, - think of it being the equivalent of many State benefits available today.
ibi