3 wire subpanel to a new 3 wire subpanel
Absolutely not.  You cannot run 3-wire to your sub-sub-panel.

You will need to run 4-wire from the grandfathered subpanel to the new sub-sub-panel.  You will need land the neutral and ground feeders on neutral and ground buses in the existing subpanel.  You will need to separate ground and neutral buses on the new panel. 

Tell you a story.  A company did a remodel on an old 2-story building.  They needed wheelchair accommodations in upstairs hallways and bathrooms. "Why?" they said, "will wheelchair users drag their wheelchairs up the stairs?" (in fact, they do).  Three years later, they built another building right next to it, with a 2nd floor walk-between.  

The other shoe will eventually drop, you will want to upsize the existing feeder.  At that point you will go 4-wire, and since the new extension is 4-wire, you are Code complete.  

Save money other ways

The current subpanel has the same breaker size as the intended feed to the sub-subpanel. As such, the $40 subpanel breaker feeding the sub-sub is redundant.  Get rid of it.  If the subpanel has feed-thru lugs, you can just tap them.  Otherwise you can get a lug "breaker" that fits in two breaker spaces but is just lugs. 

The sub-sub does not need a main breaker.  

AA-8000 Aluminum wire is your friend in feeders (not least, the lugs you'll be attaching to are aluminum, why create a dissimilar metals issue when you're paid not to).  You should use 1-1-1-3 Al instead of 3-3-3-5 Cu, so your cable cost will be $100 instead of $250.

3-wire aluminum cable in these sizes is likely AA-1350 alloy, which was outlawed except for service drops/laterals which are outdoors.

I already bought the cable. --->> At these large sizes, most of the cable's value is mineral value.  Strip the insulation off and take it to the metal recycler.