Approach the admin counter and ask for your medical reports to be transferred over to your preferred hospital. Pls also get a copy of all your reports and scans and whatever nots in case the internal transfer is unsuccessful. There may be a fee to retrieve all medical reports and may also take a while so you might want to be prepared for all that.
My friend switched from KKH to PEH and when she asked the counter to transfer all her medical reports, they said they would do it internally however it didn’t turn out that way and she had to frantically get her reports at the last min.
So be sure you make yourself clear about this with KKH admin or preferably your KKH doc. All of them say different things, it’s frustrating.