When doing a logistic regression, $y = x_1+x_2$, it gave an odds ratio for $x_1$ of $3.1$. When doing another logistic regression, $y = x_1+x_2+x_3$, it gave an odds ratio for $x_1$ of $5.1$. Does this make sense? I thought when increasing the number of adjusted variables, odds ratio for $x_1$ should be reduced. Any suggestions for interpreting why odds ratio for $x_1$ got increased after adding another covariate?
There is no particular reason why the odds ratio for one variable should go down (or up) when you add another variable. It depends on the relationships among all the variables.
For example, if you are measuring the risk of having HIV/AIDS in a general population and you have sexual orientation as one independent variable, the odds ratio will go up if you add sex as another independent variable.
Or suppose you are measuring the chance of someone being a professional athlete. If you have BMI as one variable, the OR will go up if you add age to the model.