PF: Could the institution hosting the outgoing phase apply its own taxation method?

Yes. Such an additional contract in the US can be for example to ensure equivalent benefits and social security coverage. Below is an example provided by the European Commission:

Example: A BE university recruits the fellow for the total duration of the action (i.e. provides the main employment contract under the action) and sends him/her to a US university. Continuing the Belgian social security during the stay in the US could be too expensive, so that the beneficiary asks the US partner organisation to conclude an additional employment contract, in order to insure the researcher in the US.