You should make the web service anonymous and handle a different type of authentication (assuming the service needs to be secured). You cant "pass" the FedAuth cookie because that lives in the browser. So unless you do the Web service call from the browser using ajax you wont be able to do it. One thing you could do is passing the original ADFS SAML token to the web service and validate it, but that wont be trivial in asmx.
<location path="WebService">
<authorization>
<allow users="*" />
</authorization>
</location>