You can relay your email from one of the companies you mentioned. You just need to little configuration on your server. That way your emails would be going through there server This actually they want you to do. But anyway if you are communicating to people on these big companies then you already loosing the point of secure emailing.
I have just mentioned how to do that on other post. External SMTP relay server for the domain
You can use the mail relay from competitor companies of the big company you have mentioned or use paid service like this one https://www.dnsexit.com/services/email-smtp-relay-outbound/ and even have free relay account for few thousand mail per month from some other third party site like this https://www.mailjet.com/pricing. But beware all your emails will go from there server. Big companies have made a simple process more complex because of there hidden motives.