In Milnor's book about characteristic classes, he mentions the following. Let $B$ be a $CW$ complex of dimension $n$, and let $E$ be a fiber bundle such that the fiber is $n-k-1$ connected, then it is "easy" to construct a section of $E$ over the $n-k$ skeleton of $B$. I'm not sure how to do this... The long exact sequence for fibrations tells me that $p:E\to B$ induces an isomorphism on the low homotopy groups, is this the way to go?
My logic says that if the base is very connected then it should be easy to construct a section because there is no place for monodromy-type phenomena to occur, but I'm not sure how to use that the fiber is very connected.
Any hints or references will be great, thank you!