Date Field Logging

Probably the best option is to have an audit log table named something like "Payment History"

I'm not sure how your tables and fields are set up, but you'd set it up to where each time a payment is made, an automation would trigger to create a new record in your "Payment History" table that would log who made the payment, what the payment was and since a new record is made when a payment is made, the "Date Created" would be your date paid. With this, you'd have a complete payment history a lot more capabilities.

I would prefer not to add another table. How can I accomplish this in my current table?

Can you better explain your use case.  Is it like you have a single invoice for $1,200 and there are payments made over 12 months @ $100 per month?

The problem with not having a separate table; is that a logging field is going to be impossible to interrogate for data that is reportable.  You can use automations to enter a value in your main record; and use that data to create a child-table record to store the payment information; or you can pre-build a table of 12 month's of records (or for the remaining calendar year); and then update the month-records when the payments are made.

I do not think you will get a reportable-solution from a single-text logging field; a two-table structure is the best way to go; even if you decide to have a table that stores a Year-value; with 12 monthly payment values in a single record.  This would still be a 2nd table.

Do not be afraid of tables and relationships!  They are what make Quick Base great!