Hello Deb_!
Being able to see newly added contacts in the Contacts application is something that should happen by design so I can certainly understand wanting to know what was going on if they weren't showing up there and the only way to access them was by swiping right. No worries though, we have some things to do that should get it taken care of and back to normal.
First thing that we need to check is to see where they're being saved to by default. To do that, go to Settings > Mail, Contacts, Calendars. Once you get there, scroll down to Contacts and it should show you a 'Default' account here. If the Default account is set to iCloud, then what we need to do next is to go to this website, on a computer, for iCloud: Sign in to iCloud. Click on the Contacts option there and see if it's listed or not.
If the contact isn't listed there, or if you're using some other account other than iCloud, the next thing that we need to do is quit all of the open applications including Contacts. To do that, follow these steps here: Force an app to close on your iPhone, iPad, or iPod touch
Once they're closed out, restart your iPhone and then check to see if it's showing there or not. This article shows how to restart it: Restart your iPhone, iPad, or iPod touch
If for some reason the newly added contacts still aren't showing up, we'd need to move into other troubleshooting steps to try to isolate where the issue is exactly. Those steps would be to create a backup of you iPhone and then restore it to factory settings so that we can test to see if new contacts show up or not. If they do then it could be restored from your backup to test it out again. Here are the articles for that just in case you need to do those steps:
How to back up your iPhone, iPad, and iPod touch
Use iTunes on your Mac or PC to restore your iPhone, iPad, or iPod to factory settings
Restore your iPhone, iPad, or iPod touch from a backup
Cheers!