SMS does have a cost and so a free SMS API that sends everywhere is impossible.
But Zeepmobile claims that. They have a catch of course.
Ads will be appended to the SMS send. To use an ad-free version, you have to subscribe to $9.95/month.
Zeepmobile is also not suitable to be used for SMS marketing. This is because to use the API, the users need to explicitly subscribe first. Developers are required to implement this subscription step.