If you want to speed up a given process, you need to find the bottleneck.
Generally speaking (since you give no details of the SSIS-Process) at each point in the operation, one of the systems components (CPU, RAM, I/O, Network) is operating at maximum speed. You need to find the component that contributes the most to your run time and then either speed that component up by replacing it with a faster component or reduce the load on it by redesigning the process.
Since you ruled out already the I/O to the user database(s), you need to look elsewhere. For a general ad hoc look, use the systems resource monitor (available through the task manager). For a deeper look, there are lots of performance counters available via perfmon.exe, both for OS (CPU, I/O), SSIS and SQL Server.
If you have reason to believe that DB-I/O is your bottleneck, try moving the tempdb to the SSD (if you generally have lots of load on the tempdb that is a good idea anyway). Instructions here.
Unless you give us some more details about the SSIS process in question, that's all I can say for now.