Without doubt, oil price rises are often to some extent exogenous. Think of a decision by OPEC to raise prices or of rising tensions in the Middle East. To the extent that such a shock is exogenous, there is little reason to worry about more than the knock-on effect on, in particular, wages.
But in many cases a rise in oil prices may also reflect increased economic activity. An oil price increase due to a greater demand for oil requires a different monetary policy reaction than a price increase caused by reduced supply. Underlying inflation cannot offer any clarity regarding that difference.