The primary meaning of the phrase to go bald-headed, followed by an object introduced by the preposition into, for or at, is to rush without care or caution.

Of American-English origin, it is first recorded in The Pious Editor’s Creed, from Melibœus-Hipponax. The Biglow Papers, edited, with an introduction, notes, glossary, and copious index, by Homer Wilbur, A. M., pastor of the First Church in Jaalam, and (prospective) member of many literary, learned and scientific societies (Cambridge, Massachusetts – 1848), by James Russell Lowell (1819-91), American author and diplomat:

I du believe in bein’ this
Or thet, ez it may happen
One way or t’other hendiest is
To ketch the people nappin’;
It aint by princerples nor men
My preudunt course is steadied,—
I scent wich pays the best, an’ then
Go into it baldheaded.

In the Introduction to Melibœus-Hipponax. The Biglow Papers. Second series (Boston, Massachusetts – 1867), James Russell Lowell explained that to go it bald-headed is based on the notion of leaving one’s hat behind in a rush of impetuosity—see footnote:

Bald-headed: “to go it bald-headed”; in great haste, as where one rushes out without his hat.

The earliest instances of British-English uses that I have found date from 1866 in the form to go in bald-headed for and are from The Sportsman (London):

– Thursday 1^st November 1866; the meaning is to develop a passion for:

What next? We hear that at the last meeting of the French Academy of Sciences a chemist, M Commaille, presented an elaborate analysis of cat’s milk, which he says possesses “immense strength and restorative qualities.” The French are “going in baldheaded” just now for eating horseflesh in Paris. Long ago, when the practice was rarer, Balzac expressed a dread that, when horse-eaters had done with the horse, they would eat the rider—there only being “the thickness of the saddle between the two!” Asses’ milk, we know, is good for invalids, and, if we may conjecture from the fact of men’s minds being notoriously affected by what they eat and drink, is largely consumed by other people besides invalids in this country. But cat’s milk only a Frenchman could stand!

– Thursday 6^th December 1866; the meaning is to attack:

“There is nothing more fallacious than figures, except it be—facts,” somebody once facetiously observed. You can twist figures into anything almost by a species of statistical “hanky-panky” as mystifying to the eyes of novices in the art of account cookery as is a juggler’s sleight of hand to yokels at a country fair. We see no reason, however, to doubt the figures recently put forth by the Doncaster Corporation, which is in a flourishing state. According to its lately issued annual statement the corporation has had an annual income for the year ending 31^st August last of 15,073l. 16s. 11d., of which 5287l. was received as rents from various estates, 2115l. from dividends upon navigation and other shares, 6446l. from the races of 1865, and the remainder of the sum from various other sources. We hear a great deal said about the great liberality of Doncaster to the turf. But when we consider, as Admiral Rous has ere now remarked, that “the races put 50,000l. into the Doncaster pockets;” that the racing community enriches the corporation, and pays all the rates and taxes; we cannot quite “go in bald-headed” for Doncaster liberality.

Note: The following phrases are based on similar notions:
– keep your hair on, which might have originally referred to pulling off one’s wig in exasperation, anger or frustration;
– keep your shirt on, from the image of taking off one’s shirt before getting into a fight.