Processing Salsa the Next Day?

I usually always freeze my vegetables and sauces, but decided that I want to try canning some this year also. I started easy, with salsa, and used the recipe that my mom and sisters have been using for years. The recipe is for open kettle salsa, with no processing. I put the jars through the dishwasher and heated the lids in water on the stovetop, and every jar sealed.

I have been eating salsa made this way since my childhood and never have had any health issues from it, but now I'm nervous about it after reading up on botulism and all of that lovely stuff. I just made the salsa last night and canned it around midnight. Can I process the jars tonight when I get home from work or would I need to re-boil the salsa and put everything into new jars with new lids before processing?

I have a glass stovetop so I can't use that for canning (my sisters cracked while processing jars last year and I just had to replace my own after dropping something on it, so I'm not taking any chances). I have a side burner on my gas grill outside though, so I could process the jars on that in a hot water bath if that would work.

Any information or suggestions would be greatly appreciated!