I’ve got a ways to go on this one.
Robert Montgomery
Δ 1950 Eye Witness as Adam Heyward
Δ 1949 Once More, My Darling as Collier Laing
√ 1948 June Bride as Carey Jackson
Δ 1948 The Saxon Charm as Matt Saxon
Δ 1947 Ride the Pink Horse as Lucky Gagin
√ 1947 Lady in the Lake as Phillip Marlowe*
√ 1945 They Were Expendable as Lt. Brickley*
Δ 1941 Unfinished Business as Tom Duncan
√ 1941 Here Comes Mr. Jordan as Joe Pendleton*
√ 1941 Rage in Heaven as Philip Monrell
√ 1941 Mr. & Mrs. Smith as David*
Δ 1940 Chamber of Horrors as Craig (Butler)
Δ 1940 Haunted Honeymoon as Peter Wimsey
√ 1940 The Earl of Chicago as Robert Kilmount
√ 1939 Fast and Loose as Joel Sloane
√ 1938 Three Loves Has Nancy as Malcolm
√ 1938 Yellow Jack as John O’Hara
Δ 1938 The First Hundred Years as David Conway
√ 1937 Live, Love and Learn as Bob Graham
√ 1937 Ever Since Eve as Freddy Matthews
√ 1937 Night Must Fall as Danny*
√ 1937 The Last of Mrs. Cheyney as Lord Arthur Dilling
√ 1936 Piccadilly Jim as James Crocker, Jr.*
√ 1936 Trouble for Two as Prince Florizel
Δ 1936 Petticoat Fever as Dascom Dinsmore
√ 1935 No More Ladies as Sheridan Warren
Δ 1935 Vanessa: Her Love Story as Benjamin Herries
√ 1935 Biography of a Bachelor Girl as Richard ‘Dickie’ Kurt
√ 1934 Forsaking All Others as Dillon ‘Dill’/’Dilly’ Todd
√ 1934 Hide-Out as Jonathan ‘Lucky’ Wilson
Δ 1934 Riptide as Tommie Trent
√ 1934 The Mystery of Mr. X as Nicholas Revel
Δ 1934 Fugitive Lovers as Paul Porter, aka Stephen Blaine
√ 1933 Night Flight as Auguste Pellerin
Δ 1933 Another Language as Victor Hallam
√ 1933 When Ladies Meet as Jimmie Lee*
Δ 1933 Made on Broadway as Jeff Bidwell
Δ 1933 Hell Below as Lieut. Thomas Knowlton, USN
√ 1932 Faithless as William ‘Bill’ Wade
√ 1932 Blondie of the Follies as Larry Belmont
Δ 1932 Letty Lynton as Hale Darrow
Δ 1932 But the Flesh Is Weak as Max Clement
Δ 1932 Lovers Courageous as Willie Smith
√ 1931 Private Lives as Elyot Chase
√ 1931 The Man in Possession as Raymond Dabney
Δ 1931 Shipmates as John Paul Jones
√ 1931 Strangers May Kiss as Steve
√ 1931 The Easiest Way as Jack ‘Johnny’ Madison
Δ 1931 Inspiration as André Montell
Δ 1930 War Nurse as Lt. Wally O’Brien
Δ 1930 Love in the Rough as Jack Kelly
√ 1930 Our Blushing Brides as Tony Jardine
√ 1930 The Sins of the Children as Nick Higginson
√ 1930 The Big House as Kent
√ 1930 The Divorcee as Don
Δ 1930 Free and Easy as Larry
√ 1929 Their Own Desire as John Douglas Cheever
Δ 1929 Untamed as Andy McAllister
√ 1929 So This Is College as Biff
Δ 1929 Three Live Ghosts as William Foster
*Worth seeking out
MacGuffin Movies 
I’d recommend The Man in Possession – if you like Private Lives you’ll love it. Montgomery is at his most gorgeous (and that’s saying a lot), the story is great and there are plenty of funny moments along quite similar lines to Private Lives. I’ve worked my way through about half of this list now and would pick along with Night Must Fall as my two favourites. Good luck!
Thanks for the advice! I’ll try to track “Man in Possession” down. Your description of him being his “most gorgeous” is all to enticing!