He didn't stick around in the bigs, though, a 3-0 record in 1962 was accompanied by an-over-5 ERA. So he was available for a Rule 5 pickup by the Senators in 1963 as noted on his Baseball-Reference page.
He finished his career with the Senators and their minors, but I always wondered about a small detour he took in 1964, where he was bounced to the minors and to Tacoma in the Giants' system. No mention about the transaction on his B-R page, so I went on the search. I found this from the Kane (PA) Republican of May 14, 1964:
“WASHINGTON Sold pitcher Howie Koplitz to Toronto of the International League and optioned catcher Ken Retzer to the same club on 24 - hour recall.”
Toronto had shared affiliation between the Senators and Braves...but this was a straight out sell to the team, not an option.
This was re-affirmed by the deal that sent Koplitz way out west. From the Post-Crescent of Appleton WI of Aug. 12,1964:
"The Leafs also sold pitcher Howie Koplitz to Tacoma of the PCL, and placed pitcher Marshall Bridges on the 30-day disabled list."
So Toronto, who held the rights to Koplitz, sold him to the Giants' minors. Koplitz was back in the Senators system the next season, belonging to Hawaii. I cannot find the deal that sent him there, but I did find the deal that brought Koplitz back to the majors. From the Corpus Christi Caller Times of May 23, 1965:
"The Senators purchased right hander Howie Koplitz from Hawaii of the Pacific Coast League and lefthander Barry Moore from York of the Eastern League They also optioned catcher Joe McCabe to Hawaii.."
Not a recall on option, a sale. So Koplitz was; sold to Toronto outright by Washington; sold to Tacoma by Toronto; went from Tacoma to Hawaii in an UNKNOWN TRANSACTION; and then sold to Washington from Hawaii.
If someone can fill in that Tacoma-Hawaii gap, let me know!