Estimation of age-varying covariate effects in hazard regression models is considered, where covariate dependence is monotone rather than proportional. It is well-known that the proportionality assumption underlying the Cox proportional hazards model does not hold in many applications. Further, the effect of a covariate is often monotone, in the sense that the lifetime (or duration) conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value. These ordered/monotone alternatives to proportionality constitute a popular model for covariate effects in the two-sample setup, and have recently been extended to the case of continuous covariates. Under these models, the ratio of hazard functions (or cumulative hazard functions) increases or decreases with duration. Ordered departures of this kind are common in applications, and the models provide useful and intuitively appealing descriptions of covariate dependence in non-proportional situations. Here, methods for estimating hazard regression models under monotone departures are proposed. In particular, it is shown how the histogram sieve estimators can be smoothed and order restricted estimation performed using biased bootstrap techniques such as adaptive bandwidth kernel estimators or data tilting. The performance of the estimators is compared using simulated data, and their use is illustrated with applications from biomedicine and economic duration data.