The short answer is "no," but this doesn't really attend to the crux of your question, just the semantics.

First, let's clarify some of the semantic issues in your question, then I'll take a swing at what I think you're really getting at.

Since the US was a colony, “higher education” referred to education for an individual (most often, until the educational tumult of the 19th century, a white male) that was above and beyond primary or grammar school. In the US, higher education (read: university education) was less concerned with helping students develop a particular set of skills or abilities (i.e., to get a job) than it was in helping students develop as righteous and thoughtful human beings. Immediately prior to and after the American Revolution, the general focus of higher education became a little more specific. While it was still concerned with students’ individual development, it was more interested in students’ individual development insofar as that development aided in the formation of a unified American republic. That is, higher education was conceived of as a citizenship project. But not just any kind of citizenship project. Higher education aimed at creating the future leaders of the unified American republic (i.e. politicians and statesmen.)

For the more rank and file folks, higher education might have been an option, but it was an option uncommonly considered or taken advantage of and it wasn’t really necessary, particularly if all one wanted was a job. There were apprenticeships or trade unions for skilled positions, but there was also an overabundance of unskilled jobs out there.

That said, many (if not most) of the US founders considered education for everybody to be of the utmost importance for the stability of a unified American republic. Thomas Jefferson is a good example of this kind of thinking. In American Education: The National Experience, Lawrence Cremin refers to Jefferson as the "patron saint" of American popular education. Jefferson’s unrealized dream was to have a boatload of primary or grammar schools for young citizens (white males). These primer schools would be local within a community. Then, there would be larger “colleges” or “academies” that would be centralized between several different communities. These colleges or academies would serve a slightly older student body for the purposes of serving as community leaders. Then, there would be larger, grander centralized universities that would serve even older students.

Jefferson’s idea was basically this: the larger, centralized university would produce the teachers of the smaller colleges/academies, which in turn would produce the teachers of the smallest primary schools. Similarly, the centralized universities would produce leaders at the national level (think Senator or Representative), the centralized colleges/academies would produce leaders at a district level (think mayor or city councilman), and the primary school would serve to merely educate citizens in a small community.

Significantly, the education each of these three schools provided also wasn’t about getting a job (with the exception of producing teachers for the next level of school down). Each level of schooling was aimed at making good citizens for a stronger republic. Even grammar schools, whose only ostensible teaching goals were the “three Rs” aimed to make citizens literate so they could read newspapers and stay up to date on significant issues (the Founders also conceived of the press as an essential element to form a strong citizenry.)

Now the only aspect of Jefferson’s plan to ever really come to fruition in his lifetime was the larger, centralized university (now the University of Virginia in Charlottesville). Similar universities (higher education) started cropping up at that time, all with the intention of educating statesmen, politicians, diplomats, etc. Leaders of unified republic. But if you squint and sort of turn your head sideways, you can sort of see the beginnings of a tripartite schooling model that is somewhat reminiscent of contemporary schooling today. Primary school = primary school; colleges/academies = high school; universities = universities. Maybe.

Starting in the 1830s, another aspect of Jefferson’s plan developed: common schools. These were community-based primary/grammar schools intended to educate young citizens in the rudiments of the “Three Rs.” Again, this wasn’t education aimed at any kind of job. Instead, it was education aimed at ensuring that the rank-and-file citizens of the US didn’t slip into a ignorant torpor. The common belief was that if you citizens didn’t have any education, they were more inclined to laziness and unproductiveness and crime. The Common School Movement started in the 1830s by Horace Mann (among others, though he is commonly credited), aimed at ensuring that didn’t happen. Historians such as Carl Kaestle (in Pillars of the Republic) indicate that it continued throughout the 19th and into the 20th century, with some (I forget who at the moment) suggesting that it has lasted up until the end of the Cold War (1990ish).

Following the CSM, another kind of “higher” education cropped up: normal schools. These were actually closer in philosophy to what Jefferson had in mind regarding colleges/academies than they are in philosophy to what we today would call a college/university. That is, “normal schools” were “higher” insofar as they were a step above common schools, but they were not “higher” in the traditional sense of “higher education” I described above. Essentially, normal schools aimed at training teachers for common schools (among other things).

Common schools (and normal schools) were all the rage in the 19th century. And with the 1862 Morrill Act (and the 1890 2nd Morrill Act), various colleges/universities were established that were a sort of hybrid between normal schools and universities. These were schools (many state schools, nearly all public) that sought to train leaders in various industries, not just leaders in the republic. So, for example, a bright son of a textile worker might attend Michigan State in order to learn how to become the manager of a textiles factory or some such. The college/university hybrids to follow the Morrill Acts were concerned both with preparing students to be leaders as well as workers.

Now, all throughout the 18th/19th century, there were high schools, but they did not manifest in a uniform manner. Some high schools aimed at preparing students to enter into a university. Other high schools aimed at preparing students to enter into a work force. Others were religious in nature.

It wasn’t until the tail-end of the 19th century that “high school” as we know it truly became a thing. High school finds its origins in 1892 based on a declaration of leading educational theorists/advocates commonly known as the "Committee of Ten": 12 years, with the last four dedicated to adolescent/young adult education.

The Committee didn’t really specify what secondary schooling (the last four years of the twelve mandatory years of pre-college/university schooling) was for, so for awhile it was sort of community dependent, as it had been prior. There was not just more of a legislative impetus for it. In addition to being sort of scattershot, the effects of the Committee’s declaration was also not that influential. That is, in the 20 years after the 1892 declaration, not many high schools operated, and not many people attended those that did.

About 20 years after the Committee of Ten's declaration, however, the US entered into the "high school movement" or "secondary school movement". The schools that cropped up in this movement emphasized practical skills that prepared students for entering the workforce, rather than conceptual skills that prepared students for the college/university (i.e. higher education). The unofficial end date for the high school movement was 1940, though by that time the basic "structure" of schooling in the US was set: primary schooling for students up until about 14, secondary/high school for students up until about 18, post-secondary school OR the workforce for folks 18+.

So, the short of it is this: in the US, higher education has historically not been equivalent to any education aimed at getting you a job. What the history suggests, though, is that education aimed at getting somebody a job has slowly developed over time, both in terms of pervasiveness and in terms of importance.

And this, I think, has lead to the semantic confusion in your question. When you ask about “higher education,” I think you mean college/university and part of what you mean when you consider college/university is its contemporary significance in helping somebody get a job. Historically, though, that’s not what “higher education” means, and so no, there has never been a time when high school was considered “higher education”.

That said, part of what I think you’re really asking about is whether there was ever a time in the history of American education when somebody didn’t need a college education in order to get a job. And the answer to that question is “yes, of course.”

An interesting undercurrent to your question is the degree to which “getting a good job” has either overtaken or infused itself with “being a good citizen” as a project of all education, from primary to post-secondary (i.e. higher). One way to think of it is this: what’s more important getting a good job or being a good citizen? Are these two educational goals mutually exclusive or co-dependent? As the pressure to get a good job has increased, has the pressure to be a good citizen decrease? And so on.