Another possibility, probably not as good as the ones proposed here but which can be helpful in some other cases, is to extract the boundaries of the interval on which the InterpolatingFunction is defined. This is done below in the line {{xmin, xmax}} = myfcn["Domain"]; (thank you @J.M. for this info). Then, a piecewise function is built, defined only on the appropriate interval.
myfcn = Interpolation[Table[{x, 2 x^3 - 5 x^4/3 + 5 x/2}, {x, 1, 1.5, 0.1}]];
{{xmin, xmax}} = myfcn["Domain"];
myfcn2[x_] = Piecewise[{{myfcn[x], xmin <= x <= xmax}}, Indeterminate];
ContourPlot[myfcn2[x t] == t, {t, 0, 5}, {x, 0, 1}, MaxRecursion -> 3]
(I copied the output from MichaelE2's answer as there is no visible difference).