New Software testing techniques?

Hi everyone!

I have been working in an agile environment for a long time now and we follow iterative development. As an agile team we are supposed to get things done quicker and in small chunks but some how I feel the amount of time taken for testing is increasingly too much( we work in healthcare, testing is key) and even then some cases are missed out eg : compatibility with previous versions.

I want to know if there are any new testing strategies or methods that you are using to help not only speed up testing but also make your testing framework solid.

Thanks!