High or low gear?
 

I have found that when climbing a hill, on my 1.2 litre 4 pot Jazz, I get much better mpg (as indicated by my ScanGauge E) by sticking to 5th gear, low rpm and throttle well open. If I select a lower gear, higher revs and a light throttle, the mpg figure drops (in 3rd it can go down to 50% of 5th gear figure).
And I am talking about lugging along at 1500 - 2000 rpm here. Obviously the throttle being almost full open is making the difference. No pumping restriction.
I am also trying to get out of the habit of coasting, for an upcoming driver assessment (I will be 70 in July and as I have to re-apply for my licence I am having my driving voluntarily checked at the same time). Since stopping coasting I have noticed a drop in mpg, even though I aim to make as much use of DFCO as possible. It is not a great deal, only about 5%, but once the assessment is over I will go back to engine on coasting.

An old Gassaver poster did the same; staying in high gear for hills.

What effect does that have on the engine, lugging and pulling a heavy load vs. drop one gear and not lug the engine under heavy load? Is the mpg gain worth the extra cost in engine stress? I can see maybe not dropping to 3rd and spinning pretty fast but wonder about 4th vs 5th and the strain on the engine that doesn't immediately appear in a mpg number.

Not really worried about the engine. I am not causing it to struggle. If it starts to struggle I change down. It is a 12 year old car with 103,000 miles on it. I am sure it has had to suffer worse than that in its time!

I think it depends upon how much load you are actually putting on the engine. Calling for a fraction of the power the engine can produce at that rpm is less stress than all the power it can.

Then rpms over 1500 aren't exactly low. That was around where the Sonic's automatic tried to keep it.

Just found this, which is basically what I am doing. Method 1 first section.
https://www.wikihow.com/Drive-Uphill

Nice find. That is how I approach a hill with an automatic.

The wiki needs updating on some bits, but those mostly can be summed up as "read the owner's manual". Most cars have hill start assist now, but how they work varies between manufacturers. On a Chevy, it works once you come to a stop on the incline, but Toyotas require the brake pedal to be depressed further for it to engage.

Then not every automatic low gear selection works to lock out the higher gears. Some can still access those high gears, but the shift points for every gear has been changed.

I have never driven a car with all these fancy gadgets (at least not that I was aware of). I was taught to do a handbrake hill start and have never had a problem with it. This is the first manual transmission car I have owned with more than 4 gears. For about the first 10 years of my driving I only drove 3 speed transmissions. I found 4 a bit of a handful to begin with!

My mum has an automatic, drives me insane, when she ascends this large hill, it switches to third and screams all the way up and refuses to change up, even when lifting off the throttle. It's very frustrating and annoying. She has a new car on order, another auto, but luckily it has the the paddles so at least when I'm driving it, I have control and I decide what gear to be in and when to be in it.

In my car, it's high power to weight ratio means I ascend every hill in 6th, and gently feather the throttle to maintain speed, but not overuse fuel.

I take it your mum has a CVT gearbox? I tried one out but found it too frenetic for my taste. I drove automatics for 25 years prior to this Jazz, but the traditional, hydraulic type.
If you ascended a hill with cruise control on it would often change down, but without cruise control all you had to do is ease back on the throttle and it would stay in the highest ratio.
Even with my low powered Jazz, most hills I can ascend in 5th (Top) and ease back as I crest the summit.