There are numerous directions to go which start moving you beyond ordinary least squares (OLS), linear regression. The universe of statistical methods is large! 

Two books that I particularly enjoyed are *Econometrics* by Hayashi and *Elements of Statistical Learning* by Hastie et. al. Looking back at your question, this books may be too advanced. But maybe not. An easier version of the latter is *An Introduction to Statistical Learning*.

  - *Econometrics* introduces a variety of methods through the lens of GMM and with an eye towards time-series econometrics.
  - *Elements of Statistical Learning* is a modern classic of the statistics, machine learning literature. It's great for opening your eyes to methods outside of traditional econometrics.

### Some examples beyond ordinary least squares...

 - [Maximum likelihood estimation (MLE)](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation)

   You must know this. It's a ubiquitous workhorse in statistics.

   If you can specify the [likelihood function](https://en.wikipedia.org/wiki/Likelihood_function) then parameters of the likelihood function can be estimated by maximizing the likelihood function. In certain special cases, (eg. linear regression with conditionally normal error terms) the OLS estimator is the MLE estimator. You've undoubtedly encountered MLE estimation before if you've estimated a [logit model](https://en.wikipedia.org/wiki/Logistic_regression).
 
   There are issues with applying MLE to economic models though. Often we know that an overall model is wrong and doesn't match the data, but we'd still like to test particular predictions.

 - [GMM](https://en.wikipedia.org/wiki/Generalized_method_of_moments) is another broad method for estimating parameters based upon moment conditions. Hayashi's book *Econometrics* develops ordinary least squares regression, instrumental variables, maximum likelihood, and other methods as special cases of GMM with different moment conditions.

   [A John Cochrane ode to GMM is here.](https://johnhcochrane.blogspot.com/2013/10/hansens-nobel.html)


 - [Matching methods](https://en.wikipedia.org/wiki/Matching_(statistics)) for estimating causal effects are common in certain areas of economics.

   The idea is to match a treated entity with an untreated entity based upon observable characteristics. A widely used technique for example is [propensity score matching](https://en.wikipedia.org/wiki/Propensity_score_matching)

 - Broader classes of linear methods (shrinkage estimators):

   The idea here is to start with ordinary least squares but then to bias coefficient estimates towards zero to reduce overfitting and improve out of sample prediction.

   - [Ridge regression](https://en.wikipedia.org/wiki/Tikhonov_regularization)
   - [Lasso](https://en.wikipedia.org/wiki/Lasso_(statistics))

### References
[Fumio, Hayashi, 2000, *Econometrics*](https://www.amazon.com/Econometrics-Fumio-Hayashi/dp/0691010188)

[Hastie, Trevor, Robert Tibshirani, Jerome Friedman, 2009, *Elements of Statistical Learning*](https://www.amazon.com/Elements-Statistical-Learning-Prediction-Statistics-ebook/dp/B00475AS2E)

[James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani, 2017, *An Introduction to Statistical Learning*](https://www.amazon.com/Introduction-Statistical-Learning-Applications-Statistics/dp/1461471370)