Is anything expected to be found at CERN with 14 trillion electron volts that wasn't found at 13? Based on this article

the accelerator has smashed together approximately 16 million billion
  protons since 2015, when it reached its current energy of 13 trillion
  electron volts. Planned improvements before the machine restarts in
  2021 will bring the energy up to 14 trillion electron volts

I'm all for improvements and there are probably limits to how fast they can get the protons moving based on the size of the circular tunnel, so I'm all for the upgrade, but it still seems to me that going from 13 to 14 is a very narrow window to find something new.    
My question is whether there's any hints or good reasons to believe that the increase to 14 trillion electron volts and/or increasing the number of proton-proton collisions is expected to discover something new, or if they've reached close to the limit of what CERN is likely to discover and perhaps start discussions of building a bigger one?
My guess is the latter (more collisions) might find something, but the increase from 13 to 14 is a narrow window and probably not.
 A: 
but it still seems to me that going from 13 to 14 is a very narrow window to find something new. 

It is normal for people when units go from GeV to TeV , just a little letter change, to see it as a small change, but the difference it makes to capabilities  can be a step function, because 1000 GeV make a TeV , and any particles we have seen up to now are counted in GeV at the moment.
Let me give you a real life example:
LEP, the e+e- collider in the same tunnel:

The collider's energy eventually topped 209 GeV in 2000.

I was involved in the almost discovery of the Higgs at the time, with the ALEPH experiment:

Near the end of the scheduled run time, data suggested tantalizing but inconclusive hints that the Higgs particle of a mass around 115 GeV might have been observed, a sort of Holy Grail of current high-energy physics. The run-time was extended for a few months, to no avail. The strength of the signal remained at 1.7 standard deviations which translates to the 91% confidence level, much less than the confidence expected by particle physicists to claim a discovery 

If the colliding energy could have gone maybe to 220  GeV  the Higgs might have been discovered early in the 2000sands, as we were really looking for it. 
Maybe the five  events we saw  were connected with the existence of the Higgs , probably not ( as the Higgs is supposed to have a width of 4MeV), but for sure an increase in beam energy would have revealed the real Higgs. 
One must not forget that standard models are just that, mathematical models , not molders of nature, and maybe nature has unexpected surprises around the corner.For example, it was better fitted with a new emergent quark model, than with the famous parton model of Feynman, dominant at that time. 

My question is whether there's any hints or good reasons to believe that the increase to 14 trillion electron volts and/or increasing the number of proton-proton collisions is expected to discover something new, 

So as I argue above, there were no hints that the real Higgs would be just 10 GeV larger in mass than LEP reached. The LHC was planned as the next window in energy for discovering what was there, with a map from theories flexible enough where it could have been at 1 TeV instead of 125. And there are a lot of predictions for supersymmetric or even GUTS predictions for heavy particles, which have not materialized, and are sought as the window of accessible masses enlarges. There is nothing to exclude that the next new resonance is only  a few hundreds of GeV away, and the higher energy will reveal it. 
That is what experimenting is about. Experimentalists should always use models as medieval maps: here there be dragons.
