On May 13, 1966, Simon & Garfunkel performed at Vanderbilt University in Nashville, TN.

They also were interview for the Student Newspaper, he Vanderbilt Hustler was published a week later :


Copyright: Vanderbilt Hustler
Where were you 55 years ago on April 11th? I was in a recordshop and buying my first 45rpm:

Simon & Garfunkel’s El Condor Pasa / Why don’t your write me was a major hit in the Netherlands. I remember playing it 100times, it drove most of the family nuts. I was a fan and hooked to the music of this great duo.
Next singles were “Cecilia / So long, Frank Lloyd Wright“, “Bridge over troubled water / Keep the cyustomer satisfied“, “Mrs Robinson / Old Friends/Bookends” and “The Boxer / Baby Driver“.
In a newspaper article (in Concord Monitor of 1977) I read that Simon and Garfunkel gave a free concert in Concord. Here’s a bit of that history.


The free concert was organised by The Timothy and Abigail B. Walker Lecture Fund with the thought
“that the popular music of today is valid music and should be heard“.
The concert was initially planned at the City Auditorium. The distribution of free tickets started in January. The requests for tickects must have been huge, so the concert was shifted to the Capitol Theatre which could house 1400 people. Not enough to house everybody it seemed.

Here’s the review by Hank Nichols in the Concord Monitor of February 20, 1967:

and a letter to the newspaper from a reader:

Here are 4 new dates that have been added to the lists of 1966 and 1967. Two that still have to be confirmed. From the Amhest set there is even a setlist online available.
1966-02-19 Amherst, University of Massachusetts, Amherst Curry Hicks Cage, MA, USA [tbc]
Setlist:
01 Leaves that are green
02 He was my brother
03 Sparrow
04 I am a rock
05 Homeward bound
06 The sound of silence
Encores:
07 Teenage moron
08 I shal be free
1966-04-23 Worcester, Memorial Auditorium, MA, USA [tbc]
1966-04-29 (or 28) Washington D.C., American University, MD, USA [during Spring Weekend]
and from 1967:
1967-01-21 Blacksburg, Virginia Tech, Burruss Hall [student Government Weekend], VA, USA

The Philadelphia Inquirer of November 9th 1969, mentions an upcoming concert of Simon & Garfunkel on November 23rd at The Miami Beach Auditorium/Convention Hall:

So far I have not been able to get it confirmed by an advertisement or a review.
here are the other 1969 shows as known to me now:
1969-05-02 Buffalo (NY), University of Buffalo [joining a Frank Zappa concert as Tom & Jerry]
1969-07-28/30 Los Angeles, USA, CBS Convention, Century Plaza Hotel
1969-10-31 Detroit, Cobo Hall (Start of the ’69 tour)
1969-11-01 Kansas Wichita State University / Henry Levitt Arena USA WI
1969-11-02 St. Louis, Kiel Auditorium
1969-11-07 Philadelphia, Philadelphia Civic Center, Convention Hall
1969-11-08 Carbondale Southern Illinois Univeristy / USA IL
1969-11-09 Oxford Miami University / Millett Hall USA OH
1969-11-14 Salt Lake City Salt Palace USA UT
1969-11-15 Long Beach USA CA
1969-11-27 New York City Carnegie Hall USA NY CBS
1969-11-28 New York City Carnegie Hall USA NY CBS