difference between dummy variable categories that weren't omitted Assume we have a categorical variable (one-hot encoded) with three or more categories. {race1, race2, ..., race-n}
To avoid the dummy variable trap, assume we omitted race1. Knowing the coefficients of race2,3,...n would help us compare each to race1.
What I'm trying to understand/figure-out is how to compare in the same model without rerunning the regression, the difference between race2 and race3.
 A: Assume your outcome variable Y is continuous and the model includes only the race factor, so that: 
Y = beta0 + beta1* race2 + beta2*race3 + ... +   
beta-n-1*race-n + error

Then: 
beta0 = mean value of Y when race is equal to  
race1

beta0 + beta1 = mean value of Y when race is   
equal to race2

beta0 + beta2 = mean value of Y when race is  
equal to race3

and so on. 
If you are interested in the difference in the mean value of Y between race3 and race2,
that will be given by:
(beta0 + beta2) - (beta0 + beta1) = beta2 - beta1 
So you can set your contrast as:
c = (0, -1, 1, 0, ..., 0)
where the length of the contrast vector is the same as the number of beta coefficients in the model.  
Comment:  
When a model includes dummy variables used to encode the effect of a categorical variable, what that really means is that the model actually consists of a series of sub-models - one sub-model for each category of that variable. To write down each sub-model, simply set all the dummy variables to zero and then set each dummy variable to 1 in turns (while setting all other dummy variables to zero). 
For the model:
Y = beta0 + beta1* race2 + beta2*race3 + ... +   
beta-n-1*race-n + error   (*), 

the race variable is categorical with n categories and the dummy variables race2, ..., race-n are used to encode its effect on Y. (The race1 dummy variable was omitted from the model, reflecting the fact that race1 is treated as a reference category.) 
Here are the n sub-models that can be derived from model (*). 
Sub-model 1 corresponds to race = race1 and is obtained by setting all dummy variables in model (*) to 0. Its equation is given by:
Y = beta0 + error 

In this sub-model, beta0 represents the mean value of Y when race = race1.
Sub-model 2 corresponds to race = race2 and is obtained by setting the dummy variable for race2 to 1 in model (*) and all other dummy variables to 0. Its equation is given by:
Y = beta0 + beta1 + error 

In this sub-model, beta0 + beta1 represents the mean value of Y when race = race2. 
...
Sub-model n corresponds to race = race-n and is obtained by setting the dummy variable for race-n to 1 in model (*) and all other dummy variables to 0. Its equation is given by:
Y = beta0 + beta-n + error 

In this sub-model, beta0 + beta-n represents the mean value of Y when race = race-n. 
The above sub-models help elucidate the interpretation of the parameters beta0, beta0 + beta1, ..., beta0 + beta2. Now we can construct differences between any of these parameters and interpret them. For example:


*

*(beta0 + beta1) - (beta0) = beta1 represents the difference in the  mean value of y among people for whom race = race2 and those for whom race = race1. 

*(beta0 + beta2) - (beta0 + beta1) = beta2 - beta1 represents the difference in the  mean value of y among people for whom race = race3 and those for whom race = race2. 

