Steady buckling (coiling) of thin falling liquid jets is sensitive to surface tension, yet an understanding of these capillary effects lags far behind what is known about surface-tension-free coiling. In experiments with submillimetric jets and ultralow flow rates, we find that the critical dispensing height H_{c} for coiling decreases with increasing flow rate, a trend opposite to that found previously for inertia-free coiling. We resolve the apparent contradiction using nonlinear numerical simulations based on slender-jet theory which show that the trend reversal is due to the strong effect of surface tension in our experiments. We use our experiments to construct a regime diagram (coiling vs stagnation flow) in the space of capillary number Ca and jet slenderness ε and find that it agrees well with fully nonlinear numerical simulations. However, it differs substantially from the analogous regime diagram determined experimentally by Le Merrer, Quéré, and Clanet [Phys. Rev. Lett. 109, 064502 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.064502] for the unsteady buckling of a compressed liquid bridge. Using linear stability analysis, we show that the differences between the two regime diagrams can be explained by a combination of shape nonuniformity and the influence of gravity.