The problem I have is that there is no evidence of the 24 month service having been done by the previous owner so I cannot rely on the manufacturer warranty.
Hence why I'd prefer to uncover issues, if there are any, while under the dealer warranty.
Contact Hyundai Customer Care (i.e. Hyundai Australia) -
IF the car was serviced at a
Hyundai authorised service centre, they should have a record of it (AFAIK) and if so you should be covered. If not it is technically, and quite possibly will be, as you say state (as those are the warranty conditions and if they haven't been kept...).
However, though not quite the same, I seem to remember reports of people getting the private warranty terms (full years and unlimited distance), even though they brought an ex-lease/hire (or used commercially) i30 - i.e. technically they shouldn't have due to the warranty conditions. You might find talking to Hyundai Care that they might, or might not, be prepared to make allowances as long as the missing service is done ASAP (of course elapsed distance and/or time might factor into that).
At the very least contact Hyundai Customer Care to update the owner details - that way Hyundai can contact you in the event of recalls/service campaigns (while yours is a GD, the FD generation had a couple - such as the brake switch and power steering on some aka MDPS). They can also advise you of anything like that outstanding on the vehicle.
Though probably more costly than an independent - I would suggest a Hyundai Service centre... I am guess you didn't purchase it from Hyundai dealer? If you did purchase it from a Hyundai dealer with service centre - I would take to a different service centre (try to make sure they aren't owned by the same principle/company group). Esp. if Customer Care has given you bad news... explain you want a service and to catch any faults within the 3months you have - often people are happy to oblige as it is 'sticking it' to a competitor (assuming service centre is part of a dealership) and possibly getting a new repeat customer.