In the first place, it is imperative that you contact your insurer first because they can either recommend a facility or advise you what hospitals they do not cover (take note that HeyMondo is the only insurer I know of that can pay your expenses upfront; no need to file for claims later). For other insurers who need you to file a claim, take note that to be properly and swiftly reimbursed by your travel insurance company, aside from notifying them ASAP, you should keep all the hospital receipts as well as your credit card statements so that you can claim these costs back afterward. Apart from the original copies, make sure that you keep photocopies of these receipts too.