It depends a lot on what you want, what software you use and what file formats you music is in.
The majority of software has support for displaying a single date, usually per album. How exactly this is stored depends on the tag format, but Picard’s date tag corresponds to what playback software usually uses for this. As you already found out Picard by default fills that with the release date.
If your software also supports an “original release date” maybe the default already works for you. But your wish to fill the date with originaldate indicates it probably doesn’t.
If you want to preserve the release date of the re-release separately nothing speaks against using a custom tag as you suggested. The question is again, what is your goal? If you want your player to show it, use a player that is flexible enough to be configured accordingly, such as foobar2000. If you just want to preserve the data in the tags, maybe to use it for later tagging and scripting that’s also ok.
A little digression about ID3v2: ID3v2.4 defines multiple fields for holding a date. For some reason most software uses the TDRC frame to display as date, and Picard writes date to this field for compatibility. But actually the spec defines this as the “recording time”.
For release date the TDRL “release time” would be more appropriate, but most players do not support this. And then there is TDOR “original release time”, where Picard puts “originaldate”. The only tag properly used, but not that widely supported.
ID3v2.3 is more simple. It has TDAT and TYER which together can hold a date for the recording. The spec is less explicit about it’s use, so here probably lies the reason why software usually used this for a release date. There is no frame explicitly for release date.
And there is TORY for the original release year (year only).