It's pretty simple to do so:
Firstly you need to create a new key for your company account.
Make sure that you save it to a file other than the default id_rsa otherwise you'll overwrite your default ssh key.
For illustration lets save it to ~/.ssh/id_rsa_alternate
ssh-keygen -t rsa -C "your-email-address"
Now open up your company account on Github and navigate through the settings to manage your ssh keys. Use the following command:
Copy and paste the output into a new key on your company Github account.
Next we add the new key to our identity:
Edit (or touch) your ssh config file at ~/.ssh/config and include a new option for authenticating using your company account:
Host github-COMPANY HostName github.com User git IdentityFile ~/.ssh/id_rsa_alternate
Now you can add a remote like this one:
git remote add origin git@github-COMPANY:Company/testing.git
Note that instead of using firstname.lastname@example.org you're using the host name you set up in your ssh config.