Why do we use one transformation to hide the structure in UOV?
The public key in Oil-Vinegar was proposed as $F\circ L$, where $L:\Bbb F^n \to \Bbb F^n$ ($n = o + v$) is an invertible linear transformation and $F:\Bbb F^n \to \Bbb F^o$ is an Oil-Vinegar map. 

I think you are suggesting that another transformation $T:\Bbb F^o \to \Bbb F^o$ is used at the end of the composition so that the public key becomes $P = T\circ F \circ L$. This is natural (and necessary!) in other similar schemes like Hidden Field Equations (HFE). However, this is not necessary in Oil-Vinegar since $F$ is uniformly random among all Oil-Vinegar maps and the composition of an Oil-Vinegar map with a linear transformation (on the left) yields another Oil-Vinegar map.

To be more precise, if you let your public key be $P = T\circ F \circ L$, then by definining $F' = T\circ F$ your scheme will have the shape $F'\circ L$ where $F'$ is an Oil-Vinegar map, so we can assume without loss of generality (or, without losing security strength) that the transformation $T$ does not exist.