The expression “get a foot in the door” comes from the days of door-to-door salesmen around 1900. A salesman would stick his foot in the door so that the door coudn’t be closed, thus allowing at least a few seconds longer to make a sales pitch.  People seeking employment often attempt to “get a foot in the door” of an employer, often by working part-time, or working on commission, or starting off work in a lesser position.
 
“A peddler of books sticks his foot In the door so as to prevent the lady of the house from closing it until she learns hla business,” a classic example of the saying, was cited in print in 1907. “Get a foot in the door” became an idiom in the 1930s. “Those were the days when ‘Fatty’ was comedy king of the world and a little guy named Chaplin was trying to get a foot in the door” was cited in 1933. “Somehow, I would get my foot in the door and try to create a place for myself” was cited in 1934.
 
   
Wiktionary: get one’s foot in the door
Etymology
Probably from a practice attributed to door-to-door salesmen of placing a foot in the opening of a prospective customer’s door, thereby preventing the person from closing the door until the conclusion of the sales pitch.
Verb
get one’s foot in the door
1. (idiomatic) To initiate contact or a relationship; to gain access, especially to an entry-level job.
 
(Oxford English Dictionary)
foot in the door   n.  (a) lit. a person’s foot inserted between a door and its frame to prevent the door from being shut, usually as a means of gaining unwanted access;  (b) fig. an initial introduction or way in to something, often as a basis for further progress; esp. in to get (also have) a (also one’s) foot in the door.
1856   G. H. Boker Plays & Poems II. 353   And he sang to his gittern of love and of war With one foot in his stirrup and one in her door.
1939   Ironwood (Mich.) Daily Globe 19 June 11/4   To get his foot in the door, John Lewis agreed to raise no objection.
1940   M. L. Cooke & P. Murray Organized Labor & Production xv. 186   On the other hand, in the same case, the union visualized the agreement as a ‘foot in the door’, giving it an opportunity to secure sufficient strength so that it would be flotsam when the ebb came along.
 
21 October 1907, New Albany (IN) Evening Tribune, pg. 3, col. 5:
If a peddler of books sticks his foot In the door so as to prevent the lady of the house from closing it until she learns hla business his discourtesy is likely to weigh more in her mind than the gentlemanly demeanor of half a dozen respectable agents.
   
5 October 1911, Rockford (IL) Morning Star, pg. 6, col. 6 ad:
Now every home needs a sewing machine, but the average woman waits until a clever solicitor puts his foot in the door and tries to sell her one.
(Hess Bros. & Co.—ed.)
 
1 October 1918, Bridgeport (CT) Telegram, pg. 7, col. 5:
About every 10 days in the year a woman will announce that she will NEVER buy another thing from an agent. And the very next day some agent will get his foot in the door and talk her into buying some useless doodad for a quarter clown and a quarter a week.
 
19 July 1933, Charleston (SC) Evening Post, “Night Life of Film Colony” by Mollie Merrick, pg. 3, col. 6:
Those were the days when “Fatty” was comedy king of the world and a little guy named Chaplin was trying to get a foot in the door.
 
28 June 1934, Beatrice (NE) Daily Sun, “The Upper Room,” pg. 4, col. 3:
If I were a young man and out of a job, I should find a newspaper which could use what I had to offer. Somehow, I would get my foot in the door and try to create a place for myself. It might not work elsewhere, but it would be wortha trial. The fellow who can create a job usually has one.
 
20 June 1936, Beckley (WV) Post-Herald, “Politics For Women Urged” (A), pg. 1, col. 1:
“Open your offices to them and incidentally remember that once the door is closed to women in business or professions or any other phase of our economic life, it will take generations to get where we are today. Get your foot in the door and keep it there.”
(Miss Vashti Jones of Zanesville, OH.—ed.)
   
25 January 1943, The Evening Star (Washington, DC), “This and That” by Charles E. Tracewell, pg. A-6, col. 5:
He is doing his best to live up to instruction A, in his salesmanship booklet:
 
“Get your foot in the door.”
 
OCLC WorldCat record
One foot in the door (the specialty selling formula).
Author: George F Taubeneck
Publisher: Detroit, Conjure House [1949]
Edition/Format:   Print book : English : [5th ed., rev.]
 
OCLC WorldCat record
A foot in the door : networking your way into the hidden job market
Author: Katharine Hansen
Publisher: Berkeley, Calif. : Ten Speed Press, ©2000.
Edition/Format:   Print book : English
 
Urban Dictionary
foot in the door
being in a situation of having a promising, though not as-of-yet confirmed, opportunity
(used mostly with reference to a job or related well-paid position in a company)
“you should drop your resume off to the dept. head, least that way you could still get your foot in the door for the next time they look for someone.”
by bungalow bill August 12, 2005
 
OCLC WorldCat record
Use your head to get your foot in the door : job search secrets no one else will tell you
Author: Harvey Mackay
Publisher: New York, N.Y. : Portfolio, 2010.
Edition/Format:   Print book : English