A Google Books search finds one instance of "the big picture" from 1904, in M. G. Cunniff, "The Agricultural Conquest of the Earth," in The World Work: The St. Louis Exposition (August 1904):
And agricultural processes are shown on a farm of seventy acres. Unlike most exhibits, this agricultural show may be seen with comprehension in almost any order — you may start where you will. But you see the big picture and learn the big facts in the Palace of Agriculture itself.
However, numerous earlier instances of the phrase occur in literal reference to a large painting or other picture under discussion. In addition, a semi-metaphorical instance of the phrase appears in "A Romance of the Sea-side" in Chambers's Journal of Popular Literature (July, 19, 1862):
It is just possible that you may also know that morsel of the long white terrace which fell to my lot and which chanced to be at an angle where a street broke into the terrace. I might begin to explain how it was that I, alone in my glory, came to be in lodgings at all, and how my landlady, having once had domestic relations with my family, was not a stranger to me ; but in the romance that I have to narrate, I am only a looker-on, not an actor, so that my movements or reasons for them are unimportant. Ought I to call it a romance, after all? There was nothing strange in it ; it was but a panel from the big picture of life, such a one as you yourself might have traced out during those months spent at the sea-side — a very quiet panel ; and I saw it principally through the window.
A different issue of the same periodical, published in the same year, has this interesting commentary on "what is familiarly termed the 'big picture.'" From "Home from the Colonies," in Chambers's Journal of Popular Literature (October 18, 1862):
One very tolerable test of the social state of a nation is the character of its popular prints. Gilray was not the last of our caricaturists who helped to extend and strengthen among foreigners the impression of our vulgarity and lack of wit. ... The few comic broadsheets which paved the way for Punch were (with one exception that all my contemporaries will recognize) ill conceived and clumsily executed. The young fellows who purchase that popular paper at the railway station on Wednesday afternoons to enliven them on their way down into the country before it reaches the eager hands of their sisters, have no idea of the treat which it affords, for they have never been without it. A quarter of a century ago, such combinations of head, and hand, and heart—of conception and skill, and good feeling—as are afforded every week in what is familiarly termed the 'big picture,' were not to be purchased for threepence, nor indeed for any money.
Evidently "the big picture," as the author of this article uses it, refers to a series of drawings ("social sketches") done principally (or exclusively) by John Leech and published in each weekly issue of Punch magazine during the middle decades of the 1800s.
John P. Carroll, "New York the Heart of the World," in Metropolitan Magazine (March 1899) uses the phrase in a thoroughly metaphorical (and modern) way:
Cheek by jowl with these market boats are the big steamers that carry passengers to New England cities and to the nearby points on Long Island and the villages and towns up the Hudson. All serve a purpose of usefulness in the making of a centre of commercial activity second to none in the world.
Other craft figure in the big picture, notably the tugs that bring the big ships up to the city when time is out of joint or tides are adverse. Then the same little helpers have their share in the railway trade and freight is transferred from depot to dock without delay or hindrance.
My impression from these early occurrences is that the phrase "the big picture" originated as term for a large drawing or painting, and then came to refer most especially to one in which a lot of things were going on, and finally gained the metaphorical meaning of a large-scale or long-term view of an activity or controversy.
Christine Ammer, The Facts on File Dictionary of Cliches, Second Edition (2006) has this comment on "the big picture":
This phrase, which implies that details will be omitted in favor of presenting a bird's-eye view, dates from the second half of the twentieth century. ... In Britain the term was used in the first half of the 1900s to describe the feature film in a movie presentation. However, British usage is now the same as the American.
But the Google Books search results I have cited indicate that Ammer's date of origin for the phrase is later than it ought to be.
Best Answer
From Hugh Rawson, A Dictionary of Euphemisms and Other Doubletalk (1981):
John Bartlett, Dictionary of Americanisms (1848) has this entry for the word, which does not indicate any connection between Gemini and Jesu domine:
A search of Early English Books Online returns one early (but quite ambiguous) potentially relevant match for "by Gemini," from Holinshed's Chronicles (1587):
It seems more likely that the sea-faring men are feeling joy as they use the constellation Gemini to steer by than that they are feeling joy by a euphemism along the lines of "jumping Jehosephat." I haven't yet found the 1664 source that Rawson alludes to.
The earliest use of jiminy as an expostulation that I've found in various searches is from Woman's Will: A Comedy, in The New British Theatre: A Selection of Original Dramas Not Yet Acted, volume 4 (1815):
On the other hand, the euphemistic use of Jiminy for Jesus seems quite strong in Henry Paul, Dashes of American Humour (1852):
Update (September 27, 2019): Early occurrences of variant spellings of 'Jiminy'
Prompted by the suggestions in Green's Dictionary of Slang (cited in user067531's very useful answer), I searched for some variant spellings of jiminy and found these early instances:
From William Wycherley, The Country-wife: a Comedy Acted at the Theatre Royal (1675):
From Aphra Behn, Sir Patient Fancy: A Comedy: As It Is Acted at the Duke's Theatre (1678):
From a 1684 translation of The Idylliums of Theocritus with Rapin's Discourse of Pastorals:
From (as noted in Green's Dictionary of Slang), Thomas D'Urfey, The Comical History of Don Quixote as it is acted at the Queens Theatre in Dorset-Garden (1694–1696):
And from Edward Ravenscroft, The Canterbury guests, or, A bargain broken a comedy : acted at the Theatre-Royal (1695):
From these early matches, it seems possible that the variants jeminy, geminy, jemminy, gimminy, and gemminy all may antedate the earliest print occurrence of the current standard form jiminy. It is also striking that the expression appears so frequently in plays of the period 1675–1695, often in lines spoken by servants or other people of common background. Evidently, the playwrights were familiar with the spoken expression but differed on how to spell it.