This is more of a general SSL certificate question rather than specific to Kubernetes.

If the containers/pods providing the HTTPS endpoint already have their SSL correctly configured and the SSL certificate you are using was purchased/generated from a known, trusted CA (like letsencrypt or any one of the known, trusted certificate companies out there) then there is no reason your other container apps that are making connections to your HTTPS endpoint serving pods would need anything special stored in them.

The only exception to this is if you have your own private CA and you've generated certificates on that internally and are installing them in your HTTPS serving containers. (Or if you are generating self-signed certs). Your pods/containers connecting to the https endpoints would then need to know about the CA certificate. Here is a stackoverflow question/answer that deals with this scenario:

How do I add a CA root certificate inside a docker image?

Lastly, there are better patterns to manage SSL in containers and container schedulers like Kubernetes. It all depends on your design/architecture.

Some general ideas:

1. Terminate SSL at a load balancer before traffic hits your pods. The load balancer then handles the traffic from itself to the pods as HTTP, and your clients terminate SSL at the Load Balancer. (This doesn't really tackle your specific use case though)
2. Use something like Hashicorp Vault as an internal CA, and use automation around this product and Kubernetes to manage certificates automatically.
3. Use something like cert-manager by jetstack to manage SSL in your kubernetes environment automatically. It can connect to a multitude of 'providers' such as letsencrypt for free SSL. https://github.com/jetstack/cert-manager

Hope that helps.