The answer is it depends on what you try to achieve. Post-LASSO OLS estimator is well understood, see e.g. . What has to be said is that you can never be sure that the selected covariates are 'true' covariates - LASSO, like any other estimator (e.g. adaptive LASSO) does mistakes. Oracle properties of adaptive LASSO are derived under very strong assumptions on the covariates - the so-called beta-min condition - which rarely, if at all, happens that the data is such in practice. That being said, the prediction should improve by using post-LASSO OLS or be at least as good as only using LASSO. This is because you reduce the shrinkage bias as you pointed out.
For inference, i.e. stating whether coefficients are significant or not, using post-LASSO OLS is not correct as the results are biased due to omitted variable(s) which LASSO kicks-out mistakenly (as already discussed).
The elastic net is much less understood from the theory point of view, but you may expect similar behavior of this estimator. Therefore, you may try post elastic net OLS and see if it improves your predictions. The same for adaptive versions, although from a theoretical point of view, they are not superior to standard LASSO/elastic net.
Hope this helps!
 Belloni, A., & Chernozhukov, V. (2013). Least squares after model selection in high-dimensional sparse models. Bernoulli, 19 (2), 521-547.