gt.geometric topology - Can you explain a step in a proof about 2-sided surfaces in 3-manifolds?

Following is an argument given by Hempel where I am unable to understand his comment about choosing a loop close enough to a surface. Can somebody please elucidate this:

Lemma: If $F$ is a compact connected surface properly embedded in a $3$-manifold $M$ and if $image(i_*:H_1(F;Z/2Z)rightarrow{H_1(M;Z/2Z)})=0$, then $F$ is 2-sided in $M$.

Proof: By regular neighborhood theory, it suffices to show that $F$ separates some connected neighborhood of $F$. If this is not the case then there is a loop $Jsubset{M}$ such that $Jcap{F}$ is a single point, with transverse intersection. We may choose $J$ close enough to $F$ so that $J$ is homologous to zero (mod 2) in $M$. This contradicts homological invariance (mod 2) of intersection numbers. QED.

Doubt: Everything else is clear to me except the bold part in the proof. I don't think we will be able to bring $J$ close to $F$ unless it already bounds a disc. So I can see the proof only in the simply connected $M$ case where after suitably perturbing the loop and using the loop theorem to bound, I can apply an ambient isotopy to bring the disc itself closer to $F$. But how do we see this for the non-simply connected $M$? A complicated loop in $M$ might go all around it.

Reference: Lemma 2.1, Chapter 2 (Heegard Splittings), 3-manifolds(book) by Hempel.