Computing $\pi_\ast(S)$ is a tedious business that to this day can only be done "by hand", i.e. by humans. The $p=2$ computation up to dimension 64 was completed by Kochman (see his SLNM book) with later corrections by Kochman/Mahowald. This was mainly (but not exclusively) based on the Atiyah-Hirzebruch spectral sequence $$H_\ast(BP,\pi_\ast(S)) => \pi_\ast(BP).$$
The available approximations to $\pi_\ast(S)$ try to decompose the problem into two steps:
computation of the approximation, e.g. the $E_2$ term of a spectral sequence.
computation of the differentials.
It's probably not suprising that step 2 requires human intervention; but often even the first step is a difficult computational challenge: for example, nobody seems to know how to compute the $E_2$-term of the Novikov spectral sequence efficiently.
Since this $E_2$-term is the cohomology of the moduli stack of one-dimensional formal groups, this problem should appeal to number theorists as well. And although number theory has a strong computational branch it seems that not much has been done here.
For $p\neq 2$ the answer is yes. It's an easy computation usinghomology decompositions in the sense of Eckmann–Hilton the well known exact sequence
$$\operatorname{Ext}(A,\pi_{n+1}(X))\hookrightarrow [M(A,n),X]\twoheadrightarrow \operatorname{Hom}(A,\pi_n(X)),$$
and the computation
$$\pi_{n+1}(M(A,n))=A\otimes\mathbb{Z}/2.$$
Here $M(A,n)$ is the Moore spectrum whose homology is the abelian group $A$ concentrated in degree $n$.
As you point out, it is easy to check that
$$H_n(M(A,s)\wedge M(B,t))=
\begin{cases}
A\otimes B,&n=s+t,\\
\operatorname{Tor}_1(A,B),&n=s+t+1,\\
0,&\text{otherwise}.
\end{cases}$$
Therefore, $M(A,s)\wedge M(B,t)$ can be obtained as the homotopy cofiber of a map
$$f\colon M(\operatorname{Tor}_1(A,B),s+t)\longrightarrow M(A\otimes B,s+t)$$
which is trivial in homology $H _{*}(f)=0$.
Suppose for instance that $A$ and $B$ are finite and do not have $2$-torsion. Then the previous short exact sequence shows that homology induces an isomorphism
$$H _{s+t}\colon [M(\operatorname{Tor}_1(A,B),s+t), M(A\otimes B,s+t)]\cong \operatorname{Hom}(\operatorname{Tor}_1(A,B),A\otimes B).$$
Therefore $f$ must be null-homotopic, so
$$M(A,s)\wedge M(B,t) \simeq
M(A\otimes B,s+t)\vee M(\operatorname{Tor}_1(A,B),s+t+1).$$
If you take $A=\mathbb{Z}/p^i$ and either $B=\mathbb{Z}/p^j$ or $B=\mathbb{Z}/p^i$ you always obtain the same thing on the right.
For $p= 2$ the answer is no. The spectrum $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ is the mapping cone of
$$4\colon M(\mathbb{Z}/2,0)\longrightarrow M(\mathbb{Z}/2,0)$$
which is knonw to be null-homotopic, actually $[M(\mathbb{Z}/2,0),M(\mathbb{Z}/2,0)]\cong\mathbb{Z}/4$ generated by the identity. Hence
$$M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)\simeq M(\mathbb{Z}/2,0)\vee M(\mathbb{Z}/2,1).$$
In particular the action of the Steenrod algebra on the mod 2 homology of $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ is trivial.
On the other hand, it is known that the Steenrod algebra mode 2 acts on the mod 2 homology of $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/2,0)$ in a non-trivial way, i.e. the first Steenrod operation sends the 0-dimensional generator to the 1-dimensional generator. Therefore $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ and $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/2,0)$ cannot be homotopy equivalent.
As for references, see Hatcher's book. This book doesn't deal with spectra but since we are working with finite-dimensional CW-complexes you can assume that we are in the stable range.
Best Answer
This is a comment, not an answer, I suppose. Just a reference to Adams and Walker "On complex Stiefel manifolds''. This follow up to Adams' "Vector fields on spheres'' directly computes the $KO$-groups of complex projective spaces (see Theorem 2.2) by the methods of VFS, which computed the complex $K$-theory of complex and real projective spaces and the real $K$-theory of real projective spaces. There are no extension problems in sight in the calculation of $KO(\mathbf{C}P^n)$.
Aside from the obvious, multiplication by $h_0$ in the ASS, method of detection of multiplication by $2$, there aren't a whole lot of systematic methods for detecting multiplication by $2$, let alone less simple extensions. Massey products/Toda brackets can help.