This is tricky. You can start a JQL search with date >= startOfMonth(-1) to get the 01.05 part, but I'm not sure how you'd go about getting the 16.05 part. You can hard code dates in the JQL like created >= '2021-05-01' and created <= '2021-05-15 23:59', but I know you don't want to do that, since you'd have to change it every day.
Here's an approach I have taken in cases like this, since you can embed JQL in links - create a local basic page on your workstation with links that you can manipulate with Javascript. Here's one I threw together for your purpose. You can save this somewhere as jira.htm, for example, and load it and click it when you need it. You could put other useful links on it as well.
<html>
<head>
<title>Jira Links</title>
<body>
<a id="last_month_issues">Jira Issues - Last Month Through Current Day</a>
<script>
function generateLink(){
var today = new Date();
var thisMonth = today.getMonth(); // 0=January
var thisDay = today.getDate();
var thisYear = today.getYear(); // Number of years since 1900 (e.g. 2021 = 121)
var lastMonth,lastMonthYear;
if(thisMonth == 0)
{
lastMonth = 11;
lastMonthYear = thisYear - 1;
}
else
{
lastMonth = thisMonth - 1;
lastMonthYear = thisYear;
}
lastMonth += 1;
lastMonthYear += 1900;
var myJira = "https://YOURJIRA"
var field = "created" // whichever field you wish to check
var link = myJira + "/issues/?jql=" + field + "%20>%3D%20%27" + lastMonthYear + "-" + String(lastMonth).padStart(2,'0') + "-01%27%20and%20" + field + "%20<%3D%20%27"+String(lastMonthYear).padStart(2,'0') + "-" + String(lastMonth).padStart(2,'0') + "-" + String(thisDay).padStart(2,'0') + "%2023%3A59%27";
document.getElementById("last_month_issues").href = link;
}
generateLink();
</script>
</body>
</html>
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