No. The date on which you can leave comes from your contract, not from your decision — and the law sets only a minimum below which that contract cannot go.
The minimum the law guarantees
The federal law on insurance contracts (VVG/LCA) grants an ordinary right of termination at the end of the third year, and then at the end of each following year, with three months' notice — even if the contract was concluded for a longer term. Life insurance is excluded.
This minimum works in one direction only. The parties may agree on earlier termination, and many supplementary policies offer an annual exit from the first year; what they cannot do is make termination harder than the law allows. This provision is among those that cannot be altered to the detriment of the policyholder.
An asymmetry that works in your favour
In insurance supplementing social health insurance, the ordinary right of termination belongs to the policyholder alone. The insurer does not have it. The same goes for the right to terminate after a partial loss has been paid out, which the insurer has elsewhere but not here.
In other words: you can leave at the dates provided for, but your insurer cannot drop you on the same terms.
A premium increase does not open the right people assume
For basic insurance, the KVG/LAMal grants an exit when a new premium is announced. Many people carry this reflex over to supplementary insurance: the VVG/LCA provides for nothing of the kind.
A right to terminate in the event of a price increase may exist — but because your general terms and conditions grant it, not because the law requires it. So you have to look for it in the contract, not assume it.
The gap the law does not close
For basic insurance, membership with the old insurer ends only once the new one has confirmed that it insures the person without interruption. The law itself prevents a gap in cover.
On the supplementary side, nothing of the sort exists. No provision makes the end of your contract conditional on another insurer's acceptance, and none lets you go back if the new application is refused. The usual caution — waiting for written acceptance before cancelling — is therefore not a rule of law: it is the practical consequence of there being no rule.
One last point is more reassuring: the law declares void any clause that would let the insurer stop periodic benefits already under way on the grounds that the contract is ending. Cancelling does not cut off a treatment already being paid for.