Thank you for your query.
As you are under Tier 5 (Exchange), you cannot extend your visa in the UK. You must return to your home country.
If your employer wants to recruit you, they then need to sponsor you under Tier 2 (General). You can find more information at:
I note paediatrics are on a shortage occupation list : https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/486107/Shortage_Occupation_List_-_November_2015.pdf at page 6.
This means that if your sponsor wants to employ you, they do not need to advertise for the job as your job is on the shortage occupation list.
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