How can I do separations in more colors than CMYK?

Some links that you might find of interest:

http://the-print-guide.blogspot.com/2009/06/hi-fi-color-8-strategies-to.html

http://the-print-guide.blogspot.com/2009/01/printing-at-dmaxx-maximizing-cmyk-gamut.html

https://www.pantone.com/articles/faq/what-is-extended-gamut

https://www.xrite.com/blog/5-tools-to-print-extended-gamut

https://www.idealliance.org/xcmyk

https://www.idealliance.org/ecg

Forget Hexachrome, it is dead.

You can create a multi-channel ICC profile with the appropriate ICC profiling software and Photoshop can separate to such a profile using the "Advanced" Convert to Profile option:

This is not something that really works well using "generic" profiles, best results would be for the printer to do a press run using supplied 7 colour profiling targets, then you would measure in the charts and create the profile. If you don't have the software, hardware or knowledge to do this, then I would recommend hiring a consultant to do this for you, so add that to the budget.

OK, if it is a practical concern, then I'd start with my previous question:

Do you have a CMYK vs. CMYKOGV quote from a commercial printer with the expertise to print such a project?

Do you really need +OGV? Perhaps only adding violet will be enough.

If you have the budget for 7 colour printing, how will this translate into a final commercial product? Will the target market be willing to pay extra, or will you absorb the cost and make less profit?

Run length may help, shorter runs would be too costly for 7 colour offset print, so you may go digital. There may be a wider CMYK gamut available in a given digital press, so it is then a matter of colour management and the best input image would be a wide gamut RGB image. This would need to be planned and tested with care and communication with the commercial printer.

Some digital presses offer 5 or more colours.

It is even possible to use non standard CMYK offset press setups to achieve a wider gamut than traditional CMYK. Don't forget screening, with FM screening may also improve colour in light pastel hues.

But in my opinion, it is all theoretical until you have quotes and are happy that the final printed output meets your vision for a "more lifelike" reproduction of the flowers.

I'd start with the money, that will dictate the technical requirements for image processing.

Is the job really going to an analogue press? Your printer may not discuss it with you as this is now mostly about price, so they could run the job digitally. If this was the case, the digital press would likely have a wider gamut than standard press CMYK, however, they probably simulate standard press CMYK as a standard target.

Is this a theoretical question or practical?

If practical, do you have a quote from a printer to produce the job as CMYK vs other methods?

You may want to consult with your print providers about what they can offer. 

If they offer anything beyond CMYK you might be best served to work in a »large« RGB Space (Adobe RGB, eciRGB v2, …) and let them handle separation. 

I think there may have been a renewed interest in such processes lately but it’s probably primarily relevant for packaging. 

https://www.printing.org/blog/7-things-you-need-to-know-about-7-color-printing

https://www.lifewire.com/4-6-8-color-process-printing-1077448

If you google the term Hexachrome it will provide details about a six-color process (adding Orange and Green) that Pantone introduced via Photoshop about ten years ago.  It was subsequently discontinued.  That leaves you with working with “touch plates” (called “kickers” by some old pros) that are modified versions of one or more of the CMYK plates and run in an additional special color or two to extend the normal four-color gamut.