Recommended transmission for a commuting bike I want to get a new transmission group for my bike and I'm quite confused, so I'd really apreciate some help from the community. Here's some context:

I own, since last October, a Giant Escape 2 (https://www.giant-bicycles.com/us/escape-2-2016) which I use for my daily 22km commuting (11km from home to work and 11km back) plus daily errants and the eventual lay-back ride with my GF or friends. It has a 3x8 Shimano M191-Altus group.
My regular route has several elevation changes and a couple of steep climbs along the way. According to Runtastic Road Bike, I have about 74m of elevation up and 50m of elevation down and I have realised that I mainly use the middle cog on the front crankset, going from the 3rd (on full stops and uphill) to the 5th gear, and the large cog of the frontal set when going faster, applying the 6th and 7th gear. I have an average speed of 20Km/h.

So, for my question, I'm wondering if I could do better with a 1x11 or 2x10 transmission set, since there's one frontal cog which I've literally never used in the 2500k that I've rode this bike. Or is it me that I'm not using it properly?
If you think I should replace my transmission, which one would you recommend? Considering that I don't want to spend any body part on the upgrade =)

Thank you very much for taking the time to read my amateurish-bad-english-nonsensical question and a big kudos to those who will reply it!
New you can't do it for less than $200 unless it is off brand (not Shimano or SRAM)

Shimano does not offer 1x11 in the lower end groups.

It would be like 1/2 the value of your bike.

Not using the smallest up front is not a big deal.  I would just keep what you have.  

You can remove the smallest and adjust the range if you want to.

Or you could swap to double crank.  Not sure if you can still use the same shifter or front derailleur.

Not seeing many used 1x11 yet.  Maybe look for a used 1x9.  I don't think there are any true (narrow wide) 1x8.