Three comments:

1) Don't delete outliers just to achieve a particular distribution. Outliers include information too.

2) I know the Wilcoxon rank-sign test is commonly seen as the default non-parametric alternative to the paired t-test, but it actually tests quite a different null hypothesis. The paired t-test tests an hypothesis that the mean difference is zero; the Wilcoxon test tests a null hypothesis that if we randomly select a pair of observations from the population, then the probability that observation A will be greater than observation B is 50%. It cannot be interpreted as a test of equal means or equal medians unless you add the auxiliary assumption that the two distributions are identical.

3) You could use a paired permutation test as an alternative, which allows you to stick to testing a null hypothesis of equal means (as for the t-test), but without the normality assumption. You probably can't do this in SPSS, but can do in R.