For all the top B- schools, you need to stick to the deadlines as these institutes follow utmost level of professionalism and deadlines are sacrosanct for them. All the applications had to be filled within the given stipulated time period barring which you would have to miss out on that institute. In the past, there have been very rare cases where the institutes like the one you asked about (S. P Jain) had reopened their applications post declaration of results. However, not to deny the fact that, yes there is a possibility wherein they can reopen the applications but only after the declaration of CAT results not after XAT results. You need to keep yourself abreast and keep checking their websites because they might open the portal either right before CAT or post CAT results but only for a very short span. So try not to miss out on that. Like the previous year, colleges like IMI opened its application but only for one or two weeks. They also release newspaper ads about their reopening of applications. So keep a check on that too.
Answering the next part whether you can go to such top B- schools without applying for them, then I am sorry to inform you that there is no such option. Without an authentic call letter via mail or online, you are not eligible for further rounds of admission process.
Till then, relax, you might get a chance to apply again but before that keep your eyes focused on CAT.
Wish you all the best!!  May the positives be your side.

No, generally for the top Institutes you have to apply before giving the exam. There is no practice of opening applications at a later stage

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