Can I trade in my old iPad after I have purchased a new one, not during the transaction for the new one?

You can trade-in qualifying devices whenever you wish, but if this is after you have purchased a new device, the trade-in value cannot be retrospectively applied to a prior purchase. In this situation, you will be offered an Apple Gift Card - the value of which can be used against purchase of future items.

Apple Trade In - Apple

Unfortunately, it doesn't work that way. The Trade-in service is provided only before payment for the order.

But you can sell your device at various sites.

Quick and simple answer? No.

Sorry.