Does elasticity make sense for different levels of prices? If not, how can we fix that?
The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as $EL = \frac{df(x)}{dx}\frac{x}{f(x)}$ for some function $f(x)$ (see Essential Mathematics for Economic Analysis by Sydsæter, Hammond &amp; StrØm - which is an undergraduate textbook used at many econ departments).
For example, for demand function given by: $$Q= a - bp$$ the elasticity will be given by:
$$\epsilon = \frac{dQ}{dp}\frac{p}{Q}= -b \frac{p}{a-bp}$$
so for $a=100$ and $b=10$ it will be:
$$\epsilon = - \frac{p}{100 - 10p}$$
So for example at price 5  dollars the elasticity (in its absolute value) will be 0.1 at price 9 dollars elasticity will be 0.9 and so on. Hence, elasticity can change when prices change and you don't need any new concepts. There is a special type of demand functions where elasticity will always be constant but then there are infinite number of possible demand functions and in most of them it will change with price.