'should better' is not grammatically correct.
You can use the phrase 'should do better' instead. For example, "He should do better on his exams next time."
We should better acknowledge that and them.
In fact, universities should better become smarter.
This should better satisfy the actual demand on the control quality.
I reckon I might should better try to get me a little bit more sleep.
He said that over time, the banks should better understand the standards the tests are measuring.
Should better accommodations be desired, there will be an additional charge.
The Continent, he said, should better coordinate its response to the sovereign debt crisis.
These other countries should better enforce existing economic sanctions, and employ other available levers of pressure.
Overall, people around the world should better understand Chinese people's habits, he said.
Clearly, one side of DirecTV's Web site should better know what the other one is doing.
But as educators, their judgments should better account for the nuances of child behavior.
