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1. If you roll a '1' on the first save for which you have the Improved version of the relevant save bonus feat, can you use your reroll on that save? Like if my first save vs baleful polymorph is a 1, but I have Improved Greater Fortitude, can I roll again?
2. How do these feats work with saves made by the DM in secret? (Improved Iron Will, I'm looking at you!) Is it used the first time a save is needed, or is it never used for these kinds of saves, or only when the die for the first save is less than 11 (or some other player-specified but unchanging value)?
1. If you roll a '1' on the first save for which you have the Improved version of the relevant save bonus feat, can you use your reroll on that save? Like if my first save vs baleful polymorph is a 1, but I have Improved Greater Fortitude, can I roll again?
If you fail a save to which you possess the relevant 'Improved' version of then you may re-roll that save. It doesn't matter if it a '1' or a '20' on the die roll, if failure would have been the result then you may re-roll.
2. How do these feats work with saves made by the DM in secret? (Improved Iron Will, I'm looking at you!) Is it used the first time a save is needed, or is it never used for these kinds of saves, or only when the die for the first save is less than 11 (or some other player-specified but unchanging value)?
You should remind your DM that if you are required to make a [save type] saving throw that you are permitted a re-roll 1/day on failed saves. At this point the DM should simply state that you've failed a save, and ask if you'd like to re-roll.
He shouldn't offer any info beyond that. Also, you could talk with the DM about which conditions will trigger the re-roll.