Sending email for free is going to be difficult for two reasons:
Everybody wants to do it, so it's not free, and
The services that are still free are favorite services for junk mailers and are generally blacklisted in spamfilters, have ads inserted in the emails, etc.
You can either continue doing what you're doing, using services like GMail that mostly work for you but fail every once in a while (as you are seeing), or suck it up and do one of the following:
Get a cheap email hosting account somewhere that has either no or very loose send quotas and rate limits (although note you may have to hunt around a little, as even many paid services impose strict quotas to prevent their services from being used by spammers).
Get a web hosting (and optionally email) account somewhere, write a web application that is responsible for sending the emails, and have your app make requests to that. That has the advantage of you not having to hard code your email account login credentials in all your apps, and also since all of the mail is going through the same point, you can queue mails and apply your own rate limiting so as not to exceed your email service quotas. If you get a package deal with hosted email too, you'll have the ability to manage many different email addresses.
If you have a reliable internet connection and an ISP that lets you run mail or web servers, find a cheap old PC somewhere, buy a domain name, and set up your own mail server (plenty of free SMTP servers out there).
Find an alternate way of delivering your data that isn't email. This of course depends on the nature of the data you are sending.
With the host I'm using now (I'm avoiding naming names because I don't want this to turn into a big ad fest), I pay $10.00/mo for their medium-grade shared hosting package, I get 500GB on a web server (w/ HTTPS support although certs are not free), a Tomcat server, Ruby on Rails support, 2500 email addresses, 50 FTP accounts, and a MySQL database w/ phpMyAdmin in a peaaar treeeee. * ahem * It's quite useful.
Like I mentioned above, you could also continue using GMail or finding other services here and there like that -- email services designed for a single average user that you are kind of kludging to work with an automated system. And that could be totally acceptable. If you don't mind losing a message here and there, then what you are doing is totally reasonable. However, if you want to avoid missing messages like that, you're going to have to switch to a service that is designed to do what you are trying to do, and most of those services will cost money. Personally, I feel like the $120/yr I pay is a bargain.