Hi @gustav.elmgren , welcome to the community!
We understand that this is a current limitation, please do give this feature request a vote here - Support more than 20 organizations in Organization Picker
For your use case, you may want to ask the user to specify the Organization name before providing their credentials (this being an option in the organization settings), or you could look at listing the organizations in your application by using the management API as per the suggestion here - Support more than 20 organizations in Organization Picker - #4 by adam.housman
Thanks!