Can I modify a Logistic Regression classifier to out put more than one class based on the probabilty?
We can't use the same data with a different label in the same model.
It will confuse the model.
What you need is multiple models e.g. OneVsRest strategy. i.e. one for each Class.
Little clarification-
What you are looking for is Multi-label, not Multi-class. Please check the internet on this. May read this SO Answer for a quick understanding.
You can implement an OneVsRest multi-label yourself but better use the Scikit-learn implementation Link.

OneVsRestClassifier can also be used for multilabel classification. To use this feature, provide an indicator matrix for the target y when calling .fit.

What it means is to encode your data in a multi-label format when calling fit().
You may use the from sklearn.preprocessing import MultiLabelBinarizer for this.
classif = OneVsRestClassifier(LogisticRegression())
classif.fit(X, Y)

Read this SO Answer for an example.
You may also use a sklearn.multioutput import MultiOutputClassifier wrapper for the same. Check this SO Answer