We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of random-coefficient AR(1) process driven by i.i.d. innovations in the domain of normal attraction of an $α$-stable distribution, $0< α\le 2$, as both $N$ and the time scale $n$ tend to infinity, possibly at a different rate. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $β> 0$, we show that, for $β< \max (α, 1)$, the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $α$, $β$ and the mutual increase rate of $N$ and $n$. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $α= 2$ to $0 < α< 2$.