We demonstrate high-efficiency, wideband-tunable, laser-ablated long-period fiber gratings that use an optical polymer overlay. Portions of the fiber cladding are periodically removed by ${\mathrm{CO}}_2$ laser pulses to induce periodic index changes for coupling the core mode into cladding modes. An optical polymer with a high thermo-optic coefficient with a dispersion distinct from that of silica is used on a deep-ablated cladding structure so that the effective indices of cladding modes become dispersive and the resonant wavelengths can be efficiently tuned. The tuning efficiency can be as high as $15.8\mathrm{nm}∕°\mathrm{C}$ , and the tuning range can be wider than $105\mathrm{nm}$ $\left(1545\text{–}1650\mathrm{nm}\right)$ .