Can I add a 240v outlet inline with a sub-panel supply circuit?
You need another subpanel

Being out of breaker spaces is an intolerable situation.  This invites all sorts of problems - you might balk on installing AFCI breakers because they won't fit; you might inadvertently move an old MWBC onto a tandem and burn your house down.  

So what you really need to do is install a subpanel right next to your main panel and start moving circuits over to that.  This time, get a large panel.  No one has ever written a post on SE saying how they're having problems because they have too many panel spaces.  Get a 30-40 space minimum. The moment of buying a service panel is the worst possible time to suddenly develop an urge to save $40. 

I recommend looking at panels of good quality (not bottom shelf big box store dreck) for which GFCI and AFCI are affordable (not CH or QO). 

Then, it becomes super easy.

As you observed, this question would be a no-brainer if you had the spare panel spaces.  Either put this new breaker in the new panel, or move enough 120V circuits to the new panel and put it in the old panel.