Why is 遞 = (辶 + (厂 + 虎))? 遞 is most commonly used, nowadays, to mean &quot;to pass&quot; -or- &quot;to give&quot;.
Here's the definition of its simplified counterpart 递 given on zisea:

①传送，传达：传～。投～。～送。～交。～眼色（以目示意）。呈～国书。
②顺着次序：～补。～变。～增。～减。～升。～降。
③古代指驿车。

I can only see 遞 (辶 + (厂 + 虎)) as the walking tiger factory, which I'm sure is quite absurd, but even 虒 is defined, on zisea, again, as

“委～”古书上说的一种似虎有角的兽。

in which case we have 遞 = 辶 + 虒, or the moving sī...

Why is 遞 (meaning to give, to pass) = (辶 + (厂 + 虎))?

Etymology?
说文解字：更易也。从辵虒聲。

Actually, 遞=辵+虒, it's a pictophonetic character which original meaning is "to alternate".

The meaning radical(形旁) 辵 carries the basic meaning "to walk one moment and stop the next".

The sound radical(声旁) 虒 only indicates the pronunciation of 遞(dì).