Were there any farmers in the south who paid whites and blacks to work for them and used no slaves?

To answer this question requires a little background. If you dig into the 1860 Census:

Looking cursorily over the returns, it appears that the fifteen slaveholding States contain 12,240,000 inhabitants, of whom 8,039,000 are whites, 251,000 free colored persons, and 3,950,000 are slaves.

Of these, about 30% of families owned slaves - this includes free colored persons who owned slaves. So yes, there were small farmers who owned no slaves but made do with hired labor (which could include hiring out slaves belonging to someone else).

Before answering the next part of your question, I'm going to point out that it was observed in the agricultural part of the census that "Land is abundant and cheap, while labor is scarce and dear"; the census also describes the relative undeveloped nature of American farming, which is gone into in rather more detail in Modernizing a Slave Economy: Modernizing a slave economy : the economic vision of the Confederate nation by John D. Majewski, which describes the long period when fields in Virginia and South Carolina were left fallow (20 years according to page 27) and "fire culture" was prevalent for converting "old fields" back to productive use, and contrasted this was irrigation projects and other improvements done in the more heavily populated northern states (Majewski 86).

The peculiar dynamics of Southern agriculture which made slavery so enormously profitable revolved around the cheapness of land, the relatively high cost of labor, and of course the cotton gin. In Inventing the Cotton Gin: Machine and Myth in Antebellum America by Angela Lakwete, she emphasizes the increased productivity of cotton with gin technology (and particularly in respect to improvements in gin technology); in a period where a worker could "hand-gin" a pound of cotton, improved mechanical gins allowed ginners to generate from 25 lbs a day for "foot gins" up to 4-500 lbs a day with duplex gins. This kind of thing severely benefited from economies of scale as production soared: the price of cotton dropped from 36.5 cents per pound in 1795 to as low as 10 cents per pound in 1860 - although prices varied considerably by quality of cotton, and because of massive increase in demand for cotton by the British. That was for a major cash crop, of course; farms and plantations were also very diverse, with many of the local fruits, vegetables, meat, and other products consumed locally or gone to local market (producing cotton was one thing, getting it to market was another matter due to the lack of railroads in the South). A good recent survey of Southern agriculture as a whole is Agriculture and the Confederacy : policy, productivity, and power in the Civil War South by R. Douglas Hurt, who pointed out the large number of small farmers in the South, outside of the large plantation system, and the various crops - cash and otherwise - and stock animals that they raised for profit. Which goes into answering your question at last:

Was it economically feasible under the slave model to pay employees or would you not make enough money to justify doing so?

It was economically feasible to be a small farmer in the South; we know this because there were a large number of small farmers, most of whom did not own slaves, and managed fine before 1860. However, the largest profits being made were by the rich plantation owners, who owned many slaves, could operate on economies of scale, and had the capital to invest in new and ever-more efficient gins - which is generally why the richest people in the south were plantation owners. Without slave labor, even the labor-saving nature of the cotton gin would not have made cotton the incredibly profitable behemoth that it was; but even without the cotton gin slave labor was still used in the South because of the demand for labor. So in general the small farmer could be economically viable, but slave labor was very essential to the Southern agricultural economy - as seen in part by the transition to sharecropping after the war and emancipation.