Publish WebApi to Subfolder

See this answer

Use the Visual Studio "Web Deploy" publish method.

Set "Site name" tolocalhost/api instead of localhost

After going through each folder, I noticed one folder that I did not check - Providers. And in there is a class called ApplicationOAuthProviders.cs. Once I changed values in that file, social logins started to redirect correctly.

There are a number of changes that need to change to publish to a subfolder; they are:

• ApplicationOAuthProvider: context.Request.Uri, "/YourSubfolder/"
• app.datamodel.js: self.userInfoUrl = "YourSubfolder/me"; (remove first slash)
• app.viewmodel.js : window.location = "/YourSubfolder/Account/Authorize?client_id=web&response_type=token&state=" + encodeURIComponent(window.location.hash);
• WebApiConfig.cs: routeTemplate: "YourSubfolder/{controller}/{id}"

Have you tried adding virtual directory in your IIS website?
I have my API as separate project and I can refer to it by using www.your-web-site/API-subfolder

To do it go to your IIS -> context menu on your website. Then:

And add alias (subfolder-name) and Physical path to your API project.
It should work