Service Providers should implement IDP discovery, i.e., allowing subjects to choose their Identity Provider from all available/desirable ones, by integrating it within their application. See the REFEDS Discovery Guide for details on why and how. ACOnet recommends using one of these Free/Libre software projects, which can be integrated with most any software or website:
Embedded IDP Discovery Demo
See SAML Demo SP, section "IDP Discovery Services", for demos of several systems plus of the "fallback" SWITCHwayf software.
Contact ACOnet for help with integrating IDP discovery into your eduID.at Service Provider.
Fallback discovery services
If all else fails you can make use of one of the central "fallback" discovery interfaces provided by ACOnet.
The SWITCHwayf software may be more familiar to subjects from ACOnet institutions since an older version had been in use at https://wayf.aco.net/ since 2007. This software still works (without its more dynamic features) when JavaScript is disabled in the web browser (though not much else on the web will work in such a setup):
Overview
Content Tools
Tasks