If a space $X$ is the union of two path connected open sets $U,V$ whose intersection $W=U \cap V$ has $n$ path components, then the natural thing is to choose a set $A$ consisting of one point in each path component of $W$; the form of the Seifert-van Kampen Theorem given in Topology and Groupoids, Section 6.7, determines the fundamental groupoid $\pi_1(X,A)$ as a pushout of groupoids, and from this one needs a bit of "combinatorial groupoid theory" to determine the various fundamental groups. Thus if $U,V$ are contractible, then the fundamental groups are free groups on $(n-1)$ generators.

What set me on the "groupoid path" in 1965-1968 was finding that it seemed that all of $1$-dimensional homotopy theory was quite naturally expressed in terms of groupoids rather than groups, yielding more powerful theorems with in some cases simpler proofs. In the case in point, there is no need for a diversion to covering spaces.

19 July, 2015: Here is a link to a small correction to an exposition of T&G on the Jordan Curve Theorem. It uses the algebra of groupoids to show that if
$$\begin{matrix}
C & \xrightarrow{b} & B \\
a \downarrow &&\downarrow\\
A & \to & G
\end{matrix}$$
is a pushout of groupoids such that $C$ is totally disconnected, i.e. is a disjoint union of groups, $G$ is connected, and $a,b$ are bijective on objects, then $G$ contains a free groupoid as a retract.

T&G is currently the only topology text in English to give this form of the van Kampen Theorem for nonconnected spaces. It is also given in this downloadable book Categories and Groupoids.

See also this mathoverflow discussion on "is there compelling evidence that two base points are better than one?".

Of course the covering space argument is useful to show the higher homotopy groups of the pseudocircle are trivial.