To construct infinite cyclic cover of knot exterior using seifert surface following the Lickorish's textbook, I think that we must choose a tubular neighborhood $T$ and a Seifert surface $F$ of the given knot $K$ such that $\partial T$ and $F$ intersect transversally and $T\cap F$ is a collar of $F$. Then how can we prove the existence of such $T$ and $F$?
(I only realized that transversality theorem assures the existence of $T$ and $F$ satisfying the first condition.)
Fix a metric on $S^3$ (or on $\mathbb{R}^3$). One approach is to choose $F$ first, and choose $T$ second, sufficiently small and nice with respect to $K$ and $F$. For example:
Suppose that $K$ is smooth with non-vanishing tangent. We project $K$ onto a generic plane to obtain a diagram. We build $F$ using Seifert's algorithm - we smooth the crossings of the diagram according to the orientation, cap off all circles with disks, and then attach half-twisted bands to obtain $F$ with $\partial F = K$.
Since there are only finitely many disks, and finitely many half-twisted bands, and since the projection was generic, we have a fairly nice local model for each disk and half-twisted band. Compactness gives us a small $\epsilon$ so that the $\epsilon$-neighbourhood $T$ of $K$ meets each disk and half-twisted band in a collection of small disks; these piece together to give a collar for $F$.