Robert Calton, a member at Belle Meade Country Club in Nashville, wrote a booklet on HH Barker that he published back in 2013. Somewhere on here there is probably a thread where he asked for info and I was happy to give him what I had from UK publications.
Firstly in terms of prior design work before he arrived in the US, Barker was a member at Huddersfield where Sandy Herd was the professional. As a former Open champion Herd got a reasonable amount of design work although I don't think he chased the work too much. He certainly wasn't in the same league as Braid, but anyway, in 1907 according to the club history Herd "staked out" an 18 hole course for Bradley Hall Golf Club. The club centenary book goes on to say that "Mr HH Barker, the Yorkshire and Irish Champion, has also seen the ground, and agrees with Herd that it is eminently playable. The promoters, therefore, have every encouragement to proceed with the work."
Calton suggests this was probably Barkers first taste of golf course design. By this time Barker was one of the leading amateurs in the country. At the end of June 1907 he was playing in the Open, partnering Harry Vardon in the first two rounds. This was probably his swan song in the amateur ranks as he quickly thereafter (June/July) accepted the post at Garden City.
He doesn't seem to have been in too much of a hurry to get there though as Calton has him arriving in New York on 29th September. Calton speculates that Walter Travis might have had something to do with Barker getting the job at Garden City although doesn't suggest how the connection came about. Was Barker at Sandwich in 1904 for the Amateur and met Travis then ?
Anyway, when Barker arrived, Travis apparently was in the process of making changes to the Garden City course which he discussed with Travis. Thereafter Barker seems to have had a number of commissions. The American Golfer of July 1909 gives his designs as follows;
The Richmond Country Club, Richmond (Va)
Waverly Golf Club, Portland Oregon
Spokane Country Club
Rumson Country Club, Seabright, NJ
Mayfield Club, Cleveland, Ohio
Bedford Springs Golf Club, PA
Columbia Golf Club, Washington DC
Arcola Country Club, Arcola, NJ
Courses he was "improving" included;
Springhaven Country Club, PA
Philmont Country Club, PA
Atlantic City Country Club, NJ
Newport Golf Club, RI
The booklet then goes on to describe Barker's career including his various visits back to UK, moves to different clubs and other design work that he did. I haven't got time at the moment to go through that but will try and do so later. I'll also see if I can find Robert Calton's email address for anyone interested in contacting him for a copy of his booklet.