A hot radiation environment, produced by maximizing laser-energy deposition into a small, high- "can," is a platform being developed for investigations of material properties under extreme conditions. In such small targets, almost doubling the laser energy results in only an incremental increase in the x-radiation flux, and almost no increase in the maximum achieved radiation temperature. That most of this additional laser energy is not deposited within the target is a direct consequence of laser-plasma interactions (LPI) outside of the target, which result in high-angle beams never entering the target late in the laser pulse. Accounting for these processes in the modeling results in quantitative agreement for the first time with experiments using very small cans. These findings have provided the scientific foundation for modifying the target geometry to mitigate the LPI and to achieve higher radiation temperatures.