Can I get my own private health insurance?

The one being provided by my employer is basic. Bare minimum. Inadequate.

I've already discussed it with them to change it. That I would even pay the different. But it's a no can do.

Searching online I'm not able to find any individual medical insurance for non-Saudis. The closest I could find was this Bupa expat insurance, but it only covers dependants.

Any point in the right direction would be greatly appreciated!

hi,
insurance grades are linked with positions in bupa. promotion letter/supporting letter is asked when upgrade is demanded with the insurance.
Your sponsor is the only one who can upgrade or issue your insurance, this is as per CCHI policy.
i suggest you view the cchi regulations, find a policy that favors your upgrade and negotiate with your sponsor.

Dear,
Any beneficiary has the full right to upgrade his benefit. Just need some required like internal HR memo, benefit addition request letter. Fir more info contact me on "ALIMOHD19811" via gmail.

Well if the employer is not doing it and other companies are not providing it; your option is to go for a policy with an international provider.  They might not have full direct billing in Saudi but you can always get the claims reimbursed easily.

Some options are:

CIGNA
Allianz
William Russell
AETNA