Clarification on an Hopcroft book DFA minimization example
It looks like you missed the paragraph for induction right before the Example 4.19 or figure 4.9 and the repetitive application of that induction.

INDUCTION: Let $p$ and $q$ be states such that for some input symbol $a,$ $\ r =\delta(p,a)$ and $s = \delta(q,a)$ are a pair of states known to be distinguishable. Then $\{p, q\}$ is a pair of distinguishable states.


That induction is all we employ to fill the table of state inequivalences besides the fact that every accepting state is distinguishable from every non-accepting state. (The fact holds because the empty string distinguishes them.) That is, we will never check strings that are more than one symbol long.
For example, to find whether $E$ and $G$ are distinguishable, we need to just check either $0$ or $1$ distinguishes them. Assuming $E$ and $F$ are distinguishable, since on input $1$, $E$ go to $F$, while $G$ goes to $E$, we know $E$ and $G$ are distinguishable.
You might ask, how do we know $E$ and $F$ are distinguishable? By the same procedure. &quot;Since $\{C, H\}$ is distinguishable, and states $E$ and $F$ go to $H$ and $C$, respectively, on input $0$&quot;.
Note that we may not be able to determine whether $E$ and $G$ are distinguishable right from the start by this procedure. We will try the pair $\{E,G\}$ again and again together with other pairs until we cannot find more distinguishable pairs. Although we may have to check the same pair for many times, the number of times is bounded by the number of distinguishable pairs and each time we will check just 2 possible distinguishing strings, $0$ and $1$. We have avoided testing &quot;all the possible strings&quot;, which are infinite.
You may want to look at another explanation of Hopcroft's algorithm such as this Wikipedia article or the excellent illustration at tutorialspoint.