Help using offset function

I have two tabs in my file: (1) a summary tab by month and (2) the data I wish to summarize.

On my data sheet (EMT Data), I have the following:

• Dates by week going across the top of the spreadsheet (each monday of the year)
• Job names going down the side of the spreadsheet
• Hours by job/date

On my summary tab, I have the first day of the month going down the side. For each month, I am trying to summarize the remaining hours in the month (including the current week). I have successfully used the formula below to do so. Basically the formula searches for the start/end dates sums everything in between.

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However, I now wish to turn my SUM() formula into a SUMIF(), thereby allowing me to apply a criteria to filter out certain jobs. Whenever I turn my formula into a SUMIF, I am coming up with some very unusual results. Can anyone help? Thank you!

Unfortunately, there are a number of other factors that prevent me from using a pivot table to do this.

1. The summary is actually a small piece of a much larger reporting schedule
2. The EMT data file is a system report and there is data sitting in columns between the months
3. Some months (but not all) end up being combined for scheduling purposes