Unfortunately no. Your private hospital insurance will only provide cover when you're admitted to hospital as an 'inpatient'. This refers to a person who has been admitted into an approved hospital or day facility, allocated a bed and then discharged following treatment.
If you do need to visit an emergency department in a private hospital though, but are not admitted after being assessed by the doctor or nurse - you'd be considered an 'outpatient', which means you would not be covered by HIF. An outpatient refers to someone who has received medical treatment in a doctor’s surgery or casualty department and has not been admitted into hospital.
As all Australian health insurers and Medicare are unable to pay benefits towards private emergency department treatment, you will be required the pay the fee in full.
If you're significantly out-of-pocket though, you may be eligible to claim a rebate under the Medicare Safety Net. For more information on this please click here: Medicare Safety Net
Please note: If you visit an emergency department in a public hospital though, you can rest assured that this would be covered by Medicare.