Instead of having another mempool which is extremely difficult to implement (to my knowledge), someone can create a website where people are able to submit their signed raw transaction and any interested miner may be interested to pick it up and confirm it. It won't stress the mempool out and it would ensure that the transaction is visible.
That's an interesting idea, and I agree it would be better than a second mempool. Maybe a miner could get advertising revenue from ads posted on the site for supporting the concept. This would provide alternative revenue, and may make free transactions sustainable.