TempDB Autogrowth - Does SQL need a restart?
As far as a restart goes, no you don't need to.  However there's several things to think about before growing TempDB, some of the main ones are:

-Resize all of them to the same size.  TempDB uses a round robin proporitional fill algorithm which will create hotspots on physical files which are smaller in size.  Paul Randal goes through how to properly do this here.

-There's logical and physical contention.  Logical usually arises when you have to few physical files to spread the IO requests too.  Check your metrics to see how fast your current disks are performing and benchmark it again after you get new disks.  Then you'll have an idea what your current workload threshold is.  Review the DMVs to use here.  Read more about detecting logical contention here.

If you don't care about all this nit picky performance info and this is a environment where root cause analysis, very high uptime, perf. expectations are not critical; you could just throw hardware at it but make sure you review the info around tempdb.
Edit: Aaron Bertrand gave some great things to check in the of another answer here, which I will copy here:
You should set trace flag 1117 globally but also add it to the startup parameters so it is persisted.