Unable to Add More than 3 emails for a New Contact

I'll try to help but have a few questions:

• You say "...now only allows...". Did something recently change on your iPhone?
• Are you viewing/editing your contacts in the iOS Contacts app or some other app like Gmail, Outlook, etc?
• Who is the online account provider you are using to store your contacts (e.g. Apple iCloud, Google, Microsoft, Yahoo, etc)?

I just checked my iPhone. I am using iCloud as my only Contacts provider (Settings > Apps > Contacts > Contacts Accounts). And as such, it's also my default (Settings > Apps > Contacts > Default Account). When using the Contacts app, I select a contact, click Edit at the top right, and then I can easily click "add email" below the existing email address. I just did so with a contact that already had four addresses. In addition, to the left of each it clearly shows the labels such as "work, home, other". Therefore, I am guessing that your issues are either related to the app that you are using or your account provider.