This a common misconception, if I understand your post correctly. Any new tyre will handle better initially, over the worn out ones taken off.
The Hankooks may be a better tyre, I don't know i've never had them fitted, but never compare the new tyre over the old. To get a good and fair comparison the entire life of both make should be taken into account. As the old ones WILL handle poorly as they wear.
Typical situation for an omega would be worn shoulders and almost untouched centre on the front. Rear will be the opposite with worn out centre and comparatively unworn shoulders. No car will handle well with tyres worn so.
Some improvement "may" be found by rotating rear tyres to the front. Depending if you want to only buy two at a time or a set. Although in this case it's a bit late for that by the sound of it.
Chris maybe i didn't make myself clear
I did own a few brand of tyres on migs lts put it like I compare them as new.
Of course this is my opinion.
Dunlop VS Hankook S1 - winner Hankook - costs 60% of the price of Dunlop but far better tyre.
Michelin Pilot Sport VS Hankook - both equaly good maybe Michelins slightly better until you look at the price of course.
Curently I have all year round GT Radial - strangely these tyres are quite ok in dry, cold and snow but in rain they are a piece of s... so best to avoid especially in UK
Also had Hankook Icebear winter tyres and these are also pretty bad so wouldn't reccomend them to anyone unless price is the only factor