The Swedish government exclusively makes decisions in extradition cases to countries outside the EU.
In short, the country that wants a person extradited needs to hand in a formal request to the Ministry of Justice.
Every extradition case is to be judged on its own individual merits. For that reason the Swedish government cannot provide a guarantee in advance that Julian Assange would not be subject to further extradition to the USA.
Once the British authorities enforce the UK Supreme Court's decision to extradite Julian Assange to Sweden, Sweden is bound by the so-called "Doctrine of Speciality" which means that Sweden cannot extradite him further to a third country, for example the USA, without permission from the UK. This means that Julian Assange would be in the same position in Sweden as he would be in the UK with regard to further extradition to a third country.