A fiber bundle consists of two things. The action of a group $G$ on a fiber $F$, as well as the transition functions. Given an open cover $U_i$ of the base manifold, there are local trivializations $\phi_i:\pi^{-1}(U_i)\cong U_i\times F$. On the overlap there are maps $\phi_i\phi_j^{-1}:U_i\cap U_j\times F\rightarrow U_i\cap U_j\times F$, which have the form $\phi_i\phi_j^{-1}(x,v)=(x,\tau_{ij}(x)\cdot v)$ (the dot is the group action specified above. The $\tau_{ij}$ are maps $\tau_ij:U_i\cap U_j\rightarrow G$ and satisfy the cocycle condition. Now it is a fact that if we have a continuous group action $G\times F\rightarrow F$ and $F$ is Hausdorff and locally compact, the group action extends to a continuous action $G\times \widehat{F}\rightarrow \widehat{F}$ of the one-point compactification $\widehat F$ of $F$. (I found proof of this fact in this generality in a paper by J. de Vries). Thus we can use the transition functions we had before, but the new action to construct a new fiber bundle: "the fiberwise one-point compactification".
For vector bundles the story is easier. The bundle I constructed above is bundle isomorphic to the unit sphere bundle of $E\times \mathbb R\rightarrow M$ (with respect to some auxiliary metric). To prove this you should think of the fiberwise stereographic projection. From this it is also easy to see that there are indeed two sections of this sphere bundle namely $x\mapsto (0,1)$ and $x\mapsto (0,-1)$.
Or you can think of unit disc bundle and collapse every sphere separately as Aaron Mazel-Gee suggested.
