I assume that you are a salaried PhD, because you can only buy extra insurance in form of "Lønsikring" if you are employed and have salary.
That said, I know that your concern is about that a PhD contract is only for a fixed-tem period, in your case 3 years.
Normally to be covered by "Lønsikring" - you must NOT know at the time of buying the insurance that you will become unemployed at some point in future.
However, the normal terms of the "Top-up" insurances ("Lønsikring") products on the market offers also coverage to fixed-term employed, but only if the time-limited work has been minimum 24 months.
We must admit that we do not know all the insurances-products on the market in detail, but the one offered by TopDanmark to the members of Akademikernes A-kasse has the following paragraph in their terms:
You are not covered by TopDanmark if:
"The insured works according to a fixed-term contract, after which the employment relationship is terminated within a period of
2 years after the date of insurance start/agreement of insurance"
So here TopDanmark seem to be in line with most of insurance terms seen on the market.
Some other companies (for instance the A-kasse 3F) have the further requirement that their Top-up only covers for fixed-term employed if:
"the contract has been renewed at least twice. And you have no reason to believe that it would not be renewed again."
We only know about one A-kasse who have directly mentioned coverage of PhD in their terms of Top-up insurance. This is MA A-kasse. MA A-kasse have included the following in their terms since 1. of june 2016:
"PhD employees who are paid salary after the AC-agreement for academics in the state, the agreement for academics in municipalities or regions, or have been a PhD employee after another agreement with a salary at least equal to the salary after the mentioned agreements, are also covered."
Link where you can read about this: https://ma-kasse.dk/bliv-medlem/skal-jeg-vaelge-loensikring/.
Conclusion: We will advice you to ask Akademikernes and/or TopDanmark directly about if PhD is covered. If you will not be covered in TopDanmark you could consider changing to MA A-kasse and buy the Lønsikring offered by them. In case you change to MA A-kasse all your earned "rights" in A-kasse and will be transfered to MA A-Kasse.
Hope it helps you,