Last year I was hit with a failure-to-pay penalty, even though I mailed the check on the deadline. When I spoke to an IRS agent, they claimed that the payment needed to be received, not postmarked, by April 15. Never heard of that before. He was kind enough to remove the penalty because it was a first time offense.

Since then, I pay electronically. Never again.

from the IRS (https://www.irs.gov/filing/individuals/when-to-file)

Your return is considered filed on time if your envelope is properly addressed, postmarked and deposited in the mail by the due date.

It's strange to me that they list them in that order... making me think they meant "addressed, stamped, and deposited", since postmarking clearly occurs later, after the others.

The IRS agent did not know the law (unlikely) or was just messing with you (likely)

You can also use specific DHL, FedEx, and UPS services: https://www.irs.gov/filing/private-delivery-services-pds

Many of the RTOs described in TFA already have extended hours on April 15 where you can drop it off up to midnight, and it would be post marked as the 15th even though the postmark stamping would be after midnight.

E-file. Pay online.

I don't believe that there is a way to do that without third-party software?

What about Free File Fillable Forms?

Not everyone qualifies. Thanks to tax software company lobbying.

https://www.irs.gov/pub/irs-utl/free_file_fillable_forms_use... says "Free File Fillable Forms has no age, income or residency restrictions".

IIRC if you have other than some basic income types, you cannot use it though.

That's also third party (run by "Free File Alliance, LLC").