Hausdorff theorem for any linear orders

Hausdorff theorem for any linear orders

I read this question:
Any good decomposition theorems for total orders?
and the answers. I like very much the Hausdorff theorem for scattered
linear order. I repeat it here :
Theorem (Hausdorff). A linear order is scattered iff its order type is in
some $S_\alpha$ ($\alpha$ an ordinal), where:
$S_0=\{0,1\}$
For $\alpha>0$, $S_\alpha$ is the smallest class obtained as follows: If
$\gamma$ is an ordinal, $I$ is $\gamma$, $\gamma^*$, or
$\gamma^*$+$\gamma$, and for each $i \in I$ the linear order $L_i$ is in
$\bigcup_{\beta < \alpha} S_\beta$, then $\sum_{i \in I} L_i \in
S_\alpha$.
I was wondering whether a similar characterization is possible for any
Linear order (say countable to make it simpler), by adding for example
$\eta$ (The order-type of the rationals) to $S_0$.