But I am not convinced that it will use funds the new incoming transaction, I think that it will use same input(s) and make one of the original outputs a bit smaller.
Electrum allows you to add new inputs for bumping the transaction fee.
You can do so, even if the new inputs come from a transaction made later than the original transaction.
Also, you can click on "Show advanced setting" when bumping the fee and select the strategy.
If you use the first option, there would be three cases:
1. If there's a change (which is sufficient) in the original transaction, electrum decrease the change.
2. If there is no change (or there's a change, but it's not sufficient) and there's another available UTXO in the wallet, electrum combines that UTXO with original input(s).
3. If there is no change (or there's a change, but it's not sufficient) and there is no other UTXO in the wallet, there is no option other than decreasing the payment amount.