This is essentially the same question as Replace mongodb binaries all at once? but with a different starting version (2.6 instead of 3.2).
The options I’ve outlined in that discussion apply to MongoDB 2.6 as well.
MongoDB 2.6 is below the minimum version for Cloud Manager Automation, but still supported by MongoDB Atlas Live Migration if you wanted to take that approach.
Since you are aiming for no or minimal downtime, you will have to manually upgrade through successive major versions (following the documented Upgrade Procedures) or use compatible automation tooling.
Regards,
Stennie