Hi @Andrew Nagy
When you list on the MLS, websites (redfin, realtor.com, zillow, etc) will automatically pull in the listing information from the MLS. The MLS rules prohibit the sellers information from displaying on the websites - however - the websites are required to display the listing agents phone or email.
What you want to do is find a flat fee listing agent that offers automatic call forwarding (full disclosure, we are a flat fee agent and offer this). What automatic call forwarding means is that when a buyer sees your home on a website (e.g. redfin) the buyer will be able to call the listing agent number and be automatically connected to the seller just by providing the phone system some basic property info (street number, zip code, or MLS number).
The advantages of automatic call forwarding are 1) It works 24 hours a day since it is software (and not humans), and 2) It connects the buyer live to the seller.
Also note that the flat fee agent should be placing your contact information as the seller directly in the Private Remarks to Agents field of the MLS, so the buyer agents know to contact the seller direct. 90% of buyers have agents, so this is important that agents know to deal direct with the seller.