hand over, deliver, substitute
遞 centers on transferring something from one place or person to another: handing over, delivering, or acting as a substitute.
遞 combines 辶 (movement) with 虒, which likely contributes the sound and may relate to an animal or hide, but the exact historical development is uncertain. The modern Japanese simplified form is 逓.
The movement radical 辶 suggests travel, and the inner part 虒 looks like a package being passed along. Picture a courier moving along a road to hand over a delivery.
For テイ, imagine a delivery person handing over a package and saying 'Take it, please!' — 'Take it' sounds like テイ and fits the hand-over meaning.