Context. The scarcity of spectroscopic data with a high signal-to-noise ratio in the interstellar medium between 20 and 100 μm has led to the development of several dust models with distinct dust properties that are poorly constrained in this broad wavelength range. Some of them require the presence of graphites, whereas others consider small amorphous or small aromatic carbon grains, with various dust sizes. Aims. We aim to constrain the dust emission in the mid- to far-infrared domain in the Large Magellanic Cloud (LMC) for the first time with the use of the Spitzer IRS and MIPS spectral energy distribution (SED) data, combined with Herschel data. We also consider ultraviolet extinction predictions derived from modeling. Methods. We selected ten regions that were observed as part of the SAGE-Spec program (PI: F. Kemper) to probe dust properties in various environments (diffuse, molecular, and ionized regions). All data were smoothed to the 40″ angular resolution before we extracted the dust emission spectra and photometric data. The SEDs were modeled with dust models available in the DustEM package, using the standard Mathis radiation field, as well as three additional radiation fields, with stellar clusters ages ranging from 4 Myr to 600 Myr. Results. Previous analyses of molecular clouds in the LMC have reasonably well reproduced the SEDs of the different phases of the clouds constructed from near- to far-infrared photometric data using the DustEM models. However, only by using spectroscopic data and by changing the dust abundances and size distributions in comparison with our Galaxy we were able to derive new constraints on the small- grain component. Standard dust models (with free dust abundances) that were used to reproduce the Galactic diffuse medium are clearly not able to reproduce the dust emission in the mid-infrared wavelength domain. This analysis shows the need of adjusting the parameters describing the dust size distribution, which shows a clearly distinct behavior depending on the type of environment. In addition, whereas the small-grain emission always seems to be negligible at long wavelengths in our Galaxy, the contribution of this small-dust component might be stronger than expected in the submillimeter to millimeter range in the LMC-averaged SED. Conclusions. The properties of the small-dust component of the LMC are clearly different from those of our Galaxy. Its abundance, which is significantly enhanced, might be the result of the shattering of large grains through strong shocks or turbulence. In addition, this grain component in the LMC systematically shows smaller grain sizes in the ionized regions than in the diffuse medium. Predictions of extinction curves show significantly distinct behaviors depending on the dust models, but they are also different from one region to the next. A comparison of model predictions with the LMC mean extinction curve shows that no model agrees satisfactorily when the Mathis radiation field is used, but a harder radiation field tends to improve the agreement.