Connection to SQL server fails after evry update of IIS hosted .net core app

We've deployed a .net core 3.1 app to IIS server. The app connects to a SQL server on another VM. After the initial deployment the app had no problems connecting to a database. However, every time we update the app database connections fail with the following exception:

Microsoft.Data.SqlClient.SqlException (0x80131904): Connection Timeout Expired.  The timeout period elapsed while attempting to consume the pre-login handshake acknowledgement.  This could be because the pre-login handshake failed or the server was unable to respond back in time.  The duration spent while attempting to connect to this server was - [Pre-Login] initialization=21010; handshake=2; 
 ---> System.ComponentModel.Win32Exception (258): The wait operation timed out.
   at Microsoft.Data.SqlClient.SqlInternalConnectionTds..ctor(DbConnectionPoolIdentity identity, SqlConnectionString connectionOptions, SqlCredential credential, Object providerInfo, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance, SqlConnectionString userConnectionOptions, SessionData reconnectSessionData, Boolean applyTransientFaultHandling, String accessToken, DbConnectionPool pool, SqlAuthenticationProviderManager sqlAuthProviderManager)
   at Microsoft.Data.SqlClient.SqlConnectionFactory.CreateConnection(DbConnectionOptions options, DbConnectionPoolKey poolKey, Object poolGroupProviderInfo, DbConnectionPool pool, DbConnection owningConnection, DbConnectionOptions userOptions)
   at Microsoft.Data.ProviderBase.DbConnectionFactory.CreatePooledConnection(DbConnectionPool pool, DbConnection owningObject, DbConnectionOptions options, DbConnectionPoolKey poolKey, DbConnectionOptions userOptions)
   at Microsoft.Data.ProviderBase.DbConnectionPool.CreateObject(DbConnection owningObject, DbConnectionOptions userOptions, DbConnectionInternal oldConnection)
   at Microsoft.Data.ProviderBase.DbConnectionPool.UserCreateRequest(DbConnection owningObject, DbConnectionOptions userOptions, DbConnectionInternal oldConnection)
   at Microsoft.Data.ProviderBase.DbConnectionPool.TryGetConnection(DbConnection owningObject, UInt32 waitForMultipleObjectsTimeout, Boolean allowCreate, Boolean onlyOneCheckConnection, DbConnectionOptions userOptions, DbConnectionInternal& connection)
   at Microsoft.Data.ProviderBase.DbConnectionPool.WaitForPendingOpen()

Restarting the Web site or restarting/resetting IIS does not help. The weird thing is that the connection starts working again an hour or longer after each update. This is very consistent.

Connecting to the database using SSMS on the same box is never affected by app updates - i.e. it always works.

When we update the app we first stop the site, copy new files into the app folder, and then start the site.

Deployment environment: Windows Server 2019 Standard, SQL Server 12.0.5223.6.

Has anyone run into a similar issue? Any suggestions on how to troubleshoot it?