The whole caution that is appropriate with supplementary insurance follows from this table. In basic insurance a switch has no consequences: you can go back at any time, because every insurer has to accept you. With supplementary insurance, a cancellation is a decision you may not be able to reverse.
Why you should not cancel your supplementary insurance first
The reason is that acceptance is not a formality. Anyone applying for supplementary insurance answers health questions. This pre-contractual duty of disclosure requires that all facts asked about and material to the risk assessment are stated truthfully and in full. The company decides on that basis.
The company can accept the application without restriction. It can accept it with a reservation for a particular condition. It can defer its decision, for instance until an ongoing investigation is complete. And it can reject it.
The reservation is often overlooked: it excludes only treatment of the condition concerned, and the remaining cover stays in place. So anyone who receives semi-private cover with a reservation because of a knee condition has the hospital comfort for everything else, but not for the knee. The cover exists, but precisely not for the thing they presumably wanted it for.
How the company will decide cannot be known in advance. What is decisive is what you state about your state of health. That is why the written confirmation is the only reliable signal, not an adviser's assessment and not an acquaintance's experience.
A common mistake happens here: the cancellation is sent because the deadline is pressing, and the application follows afterwards. In advisory practice that is the order that cannot be corrected after the event. Anyone who cancels first and applies afterwards can fall into a gap that can no longer be closed. A state of health that is unremarkable today can lead to a reservation in five years. So: obtain the new company's written confirmation first, check it for unwanted reservations, and only then cancel your existing cover.