although I would encourage everybody to use Bayesian statistics, calculating Bayes Factors etc. (e.g. the program 'jasp' is designed for this, and is actually easier to use than SPSS and for free), your question well has an answer: you can transform F values from an ANOVA (for main effects) to d-values. Meaning, if your regression has categorical predictors (factors) with two levels (i.e. there is a difference in two means, not more), your regression will give equal results as an ANOVA. Here is an online tool which does this F to d transformation, given the necessary information it asks for:
For categorical predictors with two levels, it would also be possible to use the t-statistics from the regression, i.e. t can be transformed to d, but then use the residual standard error as variance term. (Please anybody correct me, it is long ago I have done something like this). However, you should be aware that this only represents d then, if there are no covariates in the regression. If there are covariates, just calculate the mean of your DV on your factors levels manually, and likewise Cohens d.
If the predictors in your regression are linear, then their unstandardized weights already are "effect sizes" which are readily interpretable for how they change your DV, like d does from another perspective, and having these weights nobody would ask about a standardized mean difference on some margins of the continuous predictor, I guess. However, if so, then it would be the mean differences at the +-1SD of the of predictor variable, which you would have to do "manually" using the regression output SPSS and the calculator tools there.