A locally compact Hausdorff paracompact space is a disjoint union of clopen spaces exhaustible by compact sets (see General topology. R.Engelking theorem 5.1.27).
My definition: A topological space is called widecompact if it can be represented as disjoint union of clopen compact sets.
Any locally compact Hausdorff widecompact space is clearly paracompact. The real line $\mathbb{R}$ shows that the converse is not true.
Is there a standard name for the class of widecompact spaces? An equivalent characterization? A paper on these spaces?
This is an elongated comment, not an answer
If $X = \bigsqcup_{i\in I} X_i$ where $X_i$ are compact, then the map given by $f:X\to I$, $f[X_i] = \{i\}$ is perfect, $I$ discrete.
Conversely if a perfect map $f:X\to I$ exists such that $I$ is discrete, then $X = \bigsqcup_{i\in I} f^{-1}(i)$.
Proposition. $X$ is widecompact iff it's preimage of discrete space by a perfect map
Thus widecompact space $\implies$ paracompact Čech-complete space, since the latter spaces are precisely the preimages of completely metrizable spaces by perfect maps [Frolik].
The reason why $\mathbb{R}$ doesn't work is that it doesn't have compact components.
Proposition. Let $X$ be locally connected space with compact components, then $X$ is widecompact.
Is any LCH paracompact Čech-complete space with compact components a widecompact space?
Not sure if this is too useful, but we can write $X$ as a disjoint union of three parts $X_1, X_2, X_3$. The first part $X_1$ is a locally connected space with compact components, for example $[0, 1]$. The second, $X_2$, is a disjoint union of compact connected sets that aren't locally connected, like the topologist's sine. The third one is a disjoint union of compact spaces with no open components, like the Cantor's set.
If we could describe classes of spaces $X_2$ and $X_3$ somehow, then a classification of widecompact spaces would follow. While spaces of type $X_2$ cannot be split any further, spaces like Cantor set, which is a finite disjoint union of copies of itself, show that $X_3$ part is the worst behaved.